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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2007 Julien Pommier
// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
// Copyright (C) 2009-2019 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
/* The exp and log functions of this file initially come from
* Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
*/
#ifndef EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H
#define EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H
// IWYU pragma: private
#include "../../InternalHeaderCheck.h"
namespace Eigen {
namespace internal {
// Creates a Scalar integer type with same bit-width.
template <typename T>
struct make_integer;
template <>
struct make_integer<float> {
typedef numext::int32_t type;
};
template <>
struct make_integer<double> {
typedef numext::int64_t type;
};
template <>
struct make_integer<half> {
typedef numext::int16_t type;
};
template <>
struct make_integer<bfloat16> {
typedef numext::int16_t type;
};
/* polevl (modified for Eigen)
*
* Evaluate polynomial
*
*
*
* SYNOPSIS:
*
* int N;
* Scalar x, y, coef[N+1];
*
* y = polevl<decltype(x), N>( x, coef);
*
*
*
* DESCRIPTION:
*
* Evaluates polynomial of degree N:
*
* 2 N
* y = C + C x + C x +...+ C x
* 0 1 2 N
*
* Coefficients are stored in reverse order:
*
* coef[0] = C , ..., coef[N] = C .
* N 0
*
* The function p1evl() assumes that coef[N] = 1.0 and is
* omitted from the array. Its calling arguments are
* otherwise the same as polevl().
*
*
* The Eigen implementation is templatized. For best speed, store
* coef as a const array (constexpr), e.g.
*
* const double coef[] = {1.0, 2.0, 3.0, ...};
*
*/
template <typename Packet, int N>
struct ppolevl {
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x,
const typename unpacket_traits<Packet>::type coeff[]) {
EIGEN_STATIC_ASSERT((N > 0), YOU_MADE_A_PROGRAMMING_MISTAKE);
return pmadd(ppolevl<Packet, N - 1>::run(x, coeff), x, pset1<Packet>(coeff[N]));
}
};
template <typename Packet>
struct ppolevl<Packet, 0> {
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x,
const typename unpacket_traits<Packet>::type coeff[]) {
EIGEN_UNUSED_VARIABLE(x);
return pset1<Packet>(coeff[0]);
}
};
/* chbevl (modified for Eigen)
*
* Evaluate Chebyshev series
*
*
*
* SYNOPSIS:
*
* int N;
* Scalar x, y, coef[N], chebevl();
*
* y = chbevl( x, coef, N );
*
*
*
* DESCRIPTION:
*
* Evaluates the series
*
* N-1
* - '
* y = > coef[i] T (x/2)
* - i
* i=0
*
* of Chebyshev polynomials Ti at argument x/2.
*
* Coefficients are stored in reverse order, i.e. the zero
* order term is last in the array. Note N is the number of
* coefficients, not the order.
*
* If coefficients are for the interval a to b, x must
* have been transformed to x -> 2(2x - b - a)/(b-a) before
* entering the routine. This maps x from (a, b) to (-1, 1),
* over which the Chebyshev polynomials are defined.
*
* If the coefficients are for the inverted interval, in
* which (a, b) is mapped to (1/b, 1/a), the transformation
* required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity,
* this becomes x -> 4a/x - 1.
*
*
*
* SPEED:
*
* Taking advantage of the recurrence properties of the
* Chebyshev polynomials, the routine requires one more
* addition per loop than evaluating a nested polynomial of
* the same degree.
*
*/
template <typename Packet, int N>
struct pchebevl {
EIGEN_DEVICE_FUNC static EIGEN_STRONG_INLINE Packet run(Packet x,
const typename unpacket_traits<Packet>::type coef[]) {
typedef typename unpacket_traits<Packet>::type Scalar;
Packet b0 = pset1<Packet>(coef[0]);
Packet b1 = pset1<Packet>(static_cast<Scalar>(0.f));
Packet b2;
for (int i = 1; i < N; i++) {
b2 = b1;
b1 = b0;
b0 = psub(pmadd(x, b1, pset1<Packet>(coef[i])), b2);
}
return pmul(pset1<Packet>(static_cast<Scalar>(0.5f)), psub(b0, b2));
}
};
template <typename Packet>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Packet pfrexp_generic_get_biased_exponent(const Packet& a) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename unpacket_traits<Packet>::integer_packet PacketI;
static constexpr int mantissa_bits = numext::numeric_limits<Scalar>::digits - 1;
return pcast<PacketI, Packet>(plogical_shift_right<mantissa_bits>(preinterpret<PacketI>(pabs(a))));
}
// Safely applies frexp, correctly handles denormals.
// Assumes IEEE floating point format.
template <typename Packet>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Packet pfrexp_generic(const Packet& a, Packet& exponent) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename make_unsigned<typename make_integer<Scalar>::type>::type ScalarUI;
static constexpr int TotalBits = sizeof(Scalar) * CHAR_BIT, MantissaBits = numext::numeric_limits<Scalar>::digits - 1,
ExponentBits = TotalBits - MantissaBits - 1;
EIGEN_CONSTEXPR ScalarUI scalar_sign_mantissa_mask =
~(((ScalarUI(1) << ExponentBits) - ScalarUI(1)) << MantissaBits); // ~0x7f800000
const Packet sign_mantissa_mask = pset1frombits<Packet>(static_cast<ScalarUI>(scalar_sign_mantissa_mask));
const Packet half = pset1<Packet>(Scalar(0.5));
const Packet zero = pzero(a);
const Packet normal_min = pset1<Packet>((numext::numeric_limits<Scalar>::min)()); // Minimum normal value, 2^-126
// To handle denormals, normalize by multiplying by 2^(int(MantissaBits)+1).
const Packet is_denormal = pcmp_lt(pabs(a), normal_min);
EIGEN_CONSTEXPR ScalarUI scalar_normalization_offset = ScalarUI(MantissaBits + 1); // 24
// The following cannot be constexpr because bfloat16(uint16_t) is not constexpr.
const Scalar scalar_normalization_factor = Scalar(ScalarUI(1) << int(scalar_normalization_offset)); // 2^24
const Packet normalization_factor = pset1<Packet>(scalar_normalization_factor);
const Packet normalized_a = pselect(is_denormal, pmul(a, normalization_factor), a);
// Determine exponent offset: -126 if normal, -126-24 if denormal
const Scalar scalar_exponent_offset = -Scalar((ScalarUI(1) << (ExponentBits - 1)) - ScalarUI(2)); // -126
Packet exponent_offset = pset1<Packet>(scalar_exponent_offset);
const Packet normalization_offset = pset1<Packet>(-Scalar(scalar_normalization_offset)); // -24
exponent_offset = pselect(is_denormal, padd(exponent_offset, normalization_offset), exponent_offset);
// Determine exponent and mantissa from normalized_a.
exponent = pfrexp_generic_get_biased_exponent(normalized_a);
// Zero, Inf and NaN return 'a' unmodified, exponent is zero
// (technically the exponent is unspecified for inf/NaN, but GCC/Clang set it to zero)
const Scalar scalar_non_finite_exponent = Scalar((ScalarUI(1) << ExponentBits) - ScalarUI(1)); // 255
const Packet non_finite_exponent = pset1<Packet>(scalar_non_finite_exponent);
const Packet is_zero_or_not_finite = por(pcmp_eq(a, zero), pcmp_eq(exponent, non_finite_exponent));
const Packet m = pselect(is_zero_or_not_finite, a, por(pand(normalized_a, sign_mantissa_mask), half));
exponent = pselect(is_zero_or_not_finite, zero, padd(exponent, exponent_offset));
return m;
}
// Safely applies ldexp, correctly handles overflows, underflows and denormals.
// Assumes IEEE floating point format.
template <typename Packet>
EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Packet pldexp_generic(const Packet& a, const Packet& exponent) {
// We want to return a * 2^exponent, allowing for all possible integer
// exponents without overflowing or underflowing in intermediate
// computations.
//
// Since 'a' and the output can be denormal, the maximum range of 'exponent'
// to consider for a float is:
// -255-23 -> 255+23
// Below -278 any finite float 'a' will become zero, and above +278 any
// finite float will become inf, including when 'a' is the smallest possible
// denormal.
//
// Unfortunately, 2^(278) cannot be represented using either one or two
// finite normal floats, so we must split the scale factor into at least
// three parts. It turns out to be faster to split 'exponent' into four
// factors, since [exponent>>2] is much faster to compute that [exponent/3].
//
// Set e = min(max(exponent, -278), 278);
// b = floor(e/4);
// out = ((((a * 2^(b)) * 2^(b)) * 2^(b)) * 2^(e-3*b))
//
// This will avoid any intermediate overflows and correctly handle 0, inf,
// NaN cases.
typedef typename unpacket_traits<Packet>::integer_packet PacketI;
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename unpacket_traits<PacketI>::type ScalarI;
static constexpr int TotalBits = sizeof(Scalar) * CHAR_BIT, MantissaBits = numext::numeric_limits<Scalar>::digits - 1,
ExponentBits = TotalBits - MantissaBits - 1;
const Packet max_exponent = pset1<Packet>(Scalar((ScalarI(1) << ExponentBits) + ScalarI(MantissaBits - 1))); // 278
const PacketI bias = pset1<PacketI>((ScalarI(1) << (ExponentBits - 1)) - ScalarI(1)); // 127
const PacketI e = pcast<Packet, PacketI>(pmin(pmax(exponent, pnegate(max_exponent)), max_exponent));
PacketI b = parithmetic_shift_right<2>(e); // floor(e/4);
Packet c = preinterpret<Packet>(plogical_shift_left<MantissaBits>(padd(b, bias))); // 2^b
Packet out = pmul(pmul(pmul(a, c), c), c); // a * 2^(3b)
b = pnmadd(pset1<PacketI>(3), b, e); // e - 3b
c = preinterpret<Packet>(plogical_shift_left<MantissaBits>(padd(b, bias))); // 2^(e-3*b)
out = pmul(out, c);
return out;
}
// Explicitly multiplies
// a * (2^e)
// clamping e to the range
// [NumTraits<Scalar>::min_exponent()-2, NumTraits<Scalar>::max_exponent()]
//
// This is approx 7x faster than pldexp_impl, but will prematurely over/underflow
// if 2^e doesn't fit into a normal floating-point Scalar.
//
// Assumes IEEE floating point format
template <typename Packet>
struct pldexp_fast_impl {
typedef typename unpacket_traits<Packet>::integer_packet PacketI;
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename unpacket_traits<PacketI>::type ScalarI;
static constexpr int TotalBits = sizeof(Scalar) * CHAR_BIT, MantissaBits = numext::numeric_limits<Scalar>::digits - 1,
ExponentBits = TotalBits - MantissaBits - 1;
static EIGEN_STRONG_INLINE EIGEN_DEVICE_FUNC Packet run(const Packet& a, const Packet& exponent) {
const Packet bias = pset1<Packet>(Scalar((ScalarI(1) << (ExponentBits - 1)) - ScalarI(1))); // 127
const Packet limit = pset1<Packet>(Scalar((ScalarI(1) << ExponentBits) - ScalarI(1))); // 255
// restrict biased exponent between 0 and 255 for float.
const PacketI e = pcast<Packet, PacketI>(pmin(pmax(padd(exponent, bias), pzero(limit)), limit)); // exponent + 127
// return a * (2^e)
return pmul(a, preinterpret<Packet>(plogical_shift_left<MantissaBits>(e)));
}
};
// Natural or base 2 logarithm.
// Computes log(x) as log(2^e * m) = C*e + log(m), where the constant C =log(2)
// and m is in the range [sqrt(1/2),sqrt(2)). In this range, the logarithm can
// be easily approximated by a polynomial centered on m=1 for stability.
// TODO(gonnet): Further reduce the interval allowing for lower-degree
// polynomial interpolants -> ... -> profit!
template <typename Packet, bool base2>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog_impl_float(const Packet _x) {
const Packet cst_1 = pset1<Packet>(1.0f);
const Packet cst_minus_inf = pset1frombits<Packet>(static_cast<Eigen::numext::uint32_t>(0xff800000u));
const Packet cst_pos_inf = pset1frombits<Packet>(static_cast<Eigen::numext::uint32_t>(0x7f800000u));
const Packet cst_cephes_SQRTHF = pset1<Packet>(0.707106781186547524f);
Packet e, x;
// extract significant in the range [0.5,1) and exponent
x = pfrexp(_x, e);
// part2: Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))
// and shift by -1. The values are then centered around 0, which improves
// the stability of the polynomial evaluation.
// if( x < SQRTHF ) {
// e -= 1;
// x = x + x - 1.0;
// } else { x = x - 1.0; }
Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);
Packet tmp = pand(x, mask);
x = psub(x, cst_1);
e = psub(e, pand(cst_1, mask));
x = padd(x, tmp);
// Polynomial coefficients for rational r(x) = p(x)/q(x)
// approximating log(1+x) on [sqrt(0.5)-1;sqrt(2)-1].
constexpr float alpha[] = {0.18256296349849254f, 1.0000000190281063f, 1.0000000190281136f};
constexpr float beta[] = {0.049616247954120038f, 0.59923249590823520f, 1.4999999999999927f, 1.0f};
Packet p = ppolevl<Packet, 2>::run(x, alpha);
p = pmul(x, p);
Packet q = ppolevl<Packet, 3>::run(x, beta);
x = pdiv(p, q);
// Add the logarithm of the exponent back to the result of the interpolation.
if (base2) {
const Packet cst_log2e = pset1<Packet>(static_cast<float>(EIGEN_LOG2E));
x = pmadd(x, cst_log2e, e);
} else {
const Packet cst_ln2 = pset1<Packet>(static_cast<float>(EIGEN_LN2));
x = pmadd(e, cst_ln2, x);
}
Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));
Packet iszero_mask = pcmp_eq(_x, pzero(_x));
Packet pos_inf_mask = pcmp_eq(_x, cst_pos_inf);
// Filter out invalid inputs, i.e.:
// - negative arg will be NAN
// - 0 will be -INF
// - +INF will be +INF
return pselect(iszero_mask, cst_minus_inf, por(pselect(pos_inf_mask, cst_pos_inf, x), invalid_mask));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog_float(const Packet _x) {
return plog_impl_float<Packet, /* base2 */ false>(_x);
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog2_float(const Packet _x) {
return plog_impl_float<Packet, /* base2 */ true>(_x);
}
/* Returns the base e (2.718...) or base 2 logarithm of x.
* The argument is separated into its exponent and fractional parts.
* The logarithm of the fraction in the interval [sqrt(1/2), sqrt(2)],
* is approximated by
*
* log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
*
* for more detail see: http://www.netlib.org/cephes/
*/
template <typename Packet, bool base2>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog_impl_double(const Packet _x) {
Packet x = _x;
const Packet cst_1 = pset1<Packet>(1.0);
const Packet cst_neg_half = pset1<Packet>(-0.5);
const Packet cst_minus_inf = pset1frombits<Packet>(static_cast<uint64_t>(0xfff0000000000000ull));
const Packet cst_pos_inf = pset1frombits<Packet>(static_cast<uint64_t>(0x7ff0000000000000ull));
// Polynomial Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x)
// 1/sqrt(2) <= x < sqrt(2)
const Packet cst_cephes_SQRTHF = pset1<Packet>(0.70710678118654752440E0);
const Packet cst_cephes_log_p0 = pset1<Packet>(1.01875663804580931796E-4);
const Packet cst_cephes_log_p1 = pset1<Packet>(4.97494994976747001425E-1);
const Packet cst_cephes_log_p2 = pset1<Packet>(4.70579119878881725854E0);
const Packet cst_cephes_log_p3 = pset1<Packet>(1.44989225341610930846E1);
const Packet cst_cephes_log_p4 = pset1<Packet>(1.79368678507819816313E1);
const Packet cst_cephes_log_p5 = pset1<Packet>(7.70838733755885391666E0);
const Packet cst_cephes_log_q0 = pset1<Packet>(1.0);
const Packet cst_cephes_log_q1 = pset1<Packet>(1.12873587189167450590E1);
const Packet cst_cephes_log_q2 = pset1<Packet>(4.52279145837532221105E1);
const Packet cst_cephes_log_q3 = pset1<Packet>(8.29875266912776603211E1);
const Packet cst_cephes_log_q4 = pset1<Packet>(7.11544750618563894466E1);
const Packet cst_cephes_log_q5 = pset1<Packet>(2.31251620126765340583E1);
Packet e;
// extract significant in the range [0.5,1) and exponent
x = pfrexp(x, e);
// Shift the inputs from the range [0.5,1) to [sqrt(1/2),sqrt(2))
// and shift by -1. The values are then centered around 0, which improves
// the stability of the polynomial evaluation.
// if( x < SQRTHF ) {
// e -= 1;
// x = x + x - 1.0;
// } else { x = x - 1.0; }
Packet mask = pcmp_lt(x, cst_cephes_SQRTHF);
Packet tmp = pand(x, mask);
x = psub(x, cst_1);
e = psub(e, pand(cst_1, mask));
x = padd(x, tmp);
Packet x2 = pmul(x, x);
Packet x3 = pmul(x2, x);
// Evaluate the polynomial approximant , probably to improve instruction-level parallelism.
// y = x - 0.5*x^2 + x^3 * polevl( x, P, 5 ) / p1evl( x, Q, 5 ) );
Packet y, y1, y_;
y = pmadd(cst_cephes_log_p0, x, cst_cephes_log_p1);
y1 = pmadd(cst_cephes_log_p3, x, cst_cephes_log_p4);
y = pmadd(y, x, cst_cephes_log_p2);
y1 = pmadd(y1, x, cst_cephes_log_p5);
y_ = pmadd(y, x3, y1);
y = pmadd(cst_cephes_log_q0, x, cst_cephes_log_q1);
y1 = pmadd(cst_cephes_log_q3, x, cst_cephes_log_q4);
y = pmadd(y, x, cst_cephes_log_q2);
y1 = pmadd(y1, x, cst_cephes_log_q5);
y = pmadd(y, x3, y1);
y_ = pmul(y_, x3);
y = pdiv(y_, y);
y = pmadd(cst_neg_half, x2, y);
x = padd(x, y);
// Add the logarithm of the exponent back to the result of the interpolation.
if (base2) {
const Packet cst_log2e = pset1<Packet>(static_cast<double>(EIGEN_LOG2E));
x = pmadd(x, cst_log2e, e);
} else {
const Packet cst_ln2 = pset1<Packet>(static_cast<double>(EIGEN_LN2));
x = pmadd(e, cst_ln2, x);
}
Packet invalid_mask = pcmp_lt_or_nan(_x, pzero(_x));
Packet iszero_mask = pcmp_eq(_x, pzero(_x));
Packet pos_inf_mask = pcmp_eq(_x, cst_pos_inf);
// Filter out invalid inputs, i.e.:
// - negative arg will be NAN
// - 0 will be -INF
// - +INF will be +INF
return pselect(iszero_mask, cst_minus_inf, por(pselect(pos_inf_mask, cst_pos_inf, x), invalid_mask));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog_double(const Packet _x) {
return plog_impl_double<Packet, /* base2 */ false>(_x);
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog2_double(const Packet _x) {
return plog_impl_double<Packet, /* base2 */ true>(_x);
}
/** \internal \returns log(1 + x) computed using W. Kahan's formula.
See: http://www.plunk.org/~hatch/rightway.php
*/
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_log1p(const Packet& x) {
typedef typename unpacket_traits<Packet>::type ScalarType;
const Packet one = pset1<Packet>(ScalarType(1));
Packet xp1 = padd(x, one);
Packet small_mask = pcmp_eq(xp1, one);
Packet log1 = plog(xp1);
Packet inf_mask = pcmp_eq(xp1, log1);
Packet log_large = pmul(x, pdiv(log1, psub(xp1, one)));
return pselect(por(small_mask, inf_mask), x, log_large);
}
/** \internal \returns exp(x)-1 computed using W. Kahan's formula.
See: http://www.plunk.org/~hatch/rightway.php
*/
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_expm1(const Packet& x) {
typedef typename unpacket_traits<Packet>::type ScalarType;
const Packet one = pset1<Packet>(ScalarType(1));
const Packet neg_one = pset1<Packet>(ScalarType(-1));
Packet u = pexp(x);
Packet one_mask = pcmp_eq(u, one);
Packet u_minus_one = psub(u, one);
Packet neg_one_mask = pcmp_eq(u_minus_one, neg_one);
Packet logu = plog(u);
// The following comparison is to catch the case where
// exp(x) = +inf. It is written in this way to avoid having
// to form the constant +inf, which depends on the packet
// type.
Packet pos_inf_mask = pcmp_eq(logu, u);
Packet expm1 = pmul(u_minus_one, pdiv(x, logu));
expm1 = pselect(pos_inf_mask, u, expm1);
return pselect(one_mask, x, pselect(neg_one_mask, neg_one, expm1));
}
// Exponential function. Works by writing "x = m*log(2) + r" where
// "m = floor(x/log(2)+1/2)" and "r" is the remainder. The result is then
// "exp(x) = 2^m*exp(r)" where exp(r) is in the range [-1,1).
// exp(r) is computed using a 6th order minimax polynomial approximation.
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexp_float(const Packet _x) {
const Packet cst_zero = pset1<Packet>(0.0f);
const Packet cst_one = pset1<Packet>(1.0f);
const Packet cst_half = pset1<Packet>(0.5f);
const Packet cst_exp_hi = pset1<Packet>(88.723f);
const Packet cst_exp_lo = pset1<Packet>(-104.f);
const Packet cst_cephes_LOG2EF = pset1<Packet>(1.44269504088896341f);
const Packet cst_p2 = pset1<Packet>(0.49999988079071044921875f);
const Packet cst_p3 = pset1<Packet>(0.16666518151760101318359375f);
const Packet cst_p4 = pset1<Packet>(4.166965186595916748046875e-2f);
const Packet cst_p5 = pset1<Packet>(8.36894474923610687255859375e-3f);
const Packet cst_p6 = pset1<Packet>(1.37449637986719608306884765625e-3f);
// Clamp x.
Packet zero_mask = pcmp_lt(_x, cst_exp_lo);
Packet x = pmin(_x, cst_exp_hi);
// Express exp(x) as exp(m*ln(2) + r), start by extracting
// m = floor(x/ln(2) + 0.5).
Packet m = pfloor(pmadd(x, cst_cephes_LOG2EF, cst_half));
// Get r = x - m*ln(2). If no FMA instructions are available, m*ln(2) is
// subtracted out in two parts, m*C1+m*C2 = m*ln(2), to avoid accumulating
// truncation errors.
const Packet cst_cephes_exp_C1 = pset1<Packet>(-0.693359375f);
const Packet cst_cephes_exp_C2 = pset1<Packet>(2.12194440e-4f);
Packet r = pmadd(m, cst_cephes_exp_C1, x);
r = pmadd(m, cst_cephes_exp_C2, r);
// Evaluate the 6th order polynomial approximation to exp(r)
// with r in the interval [-ln(2)/2;ln(2)/2].
const Packet r2 = pmul(r, r);
Packet p_even = pmadd(r2, cst_p6, cst_p4);
const Packet p_odd = pmadd(r2, cst_p5, cst_p3);
p_even = pmadd(r2, p_even, cst_p2);
const Packet p_low = padd(r, cst_one);
Packet y = pmadd(r, p_odd, p_even);
y = pmadd(r2, y, p_low);
// Return 2^m * exp(r).
// TODO: replace pldexp with faster implementation since y in [-1, 1).
return pselect(zero_mask, cst_zero, pmax(pldexp(y, m), _x));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexp_double(const Packet _x) {
Packet x = _x;
const Packet cst_zero = pset1<Packet>(0.0);
const Packet cst_1 = pset1<Packet>(1.0);
const Packet cst_2 = pset1<Packet>(2.0);
const Packet cst_half = pset1<Packet>(0.5);
const Packet cst_exp_hi = pset1<Packet>(709.784);
const Packet cst_exp_lo = pset1<Packet>(-709.784);
const Packet cst_cephes_LOG2EF = pset1<Packet>(1.4426950408889634073599);
const Packet cst_cephes_exp_p0 = pset1<Packet>(1.26177193074810590878e-4);
const Packet cst_cephes_exp_p1 = pset1<Packet>(3.02994407707441961300e-2);
const Packet cst_cephes_exp_p2 = pset1<Packet>(9.99999999999999999910e-1);
const Packet cst_cephes_exp_q0 = pset1<Packet>(3.00198505138664455042e-6);
const Packet cst_cephes_exp_q1 = pset1<Packet>(2.52448340349684104192e-3);
const Packet cst_cephes_exp_q2 = pset1<Packet>(2.27265548208155028766e-1);
const Packet cst_cephes_exp_q3 = pset1<Packet>(2.00000000000000000009e0);
const Packet cst_cephes_exp_C1 = pset1<Packet>(0.693145751953125);
const Packet cst_cephes_exp_C2 = pset1<Packet>(1.42860682030941723212e-6);
Packet tmp, fx;
// clamp x
Packet zero_mask = pcmp_lt(_x, cst_exp_lo);
x = pmin(x, cst_exp_hi);
// Express exp(x) as exp(g + n*log(2)).
fx = pmadd(cst_cephes_LOG2EF, x, cst_half);
// Get the integer modulus of log(2), i.e. the "n" described above.
fx = pfloor(fx);
// Get the remainder modulo log(2), i.e. the "g" described above. Subtract
// n*log(2) out in two steps, i.e. n*C1 + n*C2, C1+C2=log2 to get the last
// digits right.
tmp = pmul(fx, cst_cephes_exp_C1);
Packet z = pmul(fx, cst_cephes_exp_C2);
x = psub(x, tmp);
x = psub(x, z);
Packet x2 = pmul(x, x);
// Evaluate the numerator polynomial of the rational interpolant.
Packet px = cst_cephes_exp_p0;
px = pmadd(px, x2, cst_cephes_exp_p1);
px = pmadd(px, x2, cst_cephes_exp_p2);
px = pmul(px, x);
// Evaluate the denominator polynomial of the rational interpolant.
Packet qx = cst_cephes_exp_q0;
qx = pmadd(qx, x2, cst_cephes_exp_q1);
qx = pmadd(qx, x2, cst_cephes_exp_q2);
qx = pmadd(qx, x2, cst_cephes_exp_q3);
// I don't really get this bit, copied from the SSE2 routines, so...
// TODO(gonnet): Figure out what is going on here, perhaps find a better
// rational interpolant?
x = pdiv(px, psub(qx, px));
x = pmadd(cst_2, x, cst_1);
// Construct the result 2^n * exp(g) = e * x. The max is used to catch
// non-finite values in the input.
// TODO: replace pldexp with faster implementation since x in [-1, 1).
return pselect(zero_mask, cst_zero, pmax(pldexp(x, fx), _x));
}
// The following code is inspired by the following stack-overflow answer:
// https://stackoverflow.com/questions/30463616/payne-hanek-algorithm-implementation-in-c/30465751#30465751
// It has been largely optimized:
// - By-pass calls to frexp.
// - Aligned loads of required 96 bits of 2/pi. This is accomplished by
// (1) balancing the mantissa and exponent to the required bits of 2/pi are
// aligned on 8-bits, and (2) replicating the storage of the bits of 2/pi.
// - Avoid a branch in rounding and extraction of the remaining fractional part.
// Overall, I measured a speed up higher than x2 on x86-64.
inline float trig_reduce_huge(float xf, Eigen::numext::int32_t* quadrant) {
using Eigen::numext::int32_t;
using Eigen::numext::int64_t;
using Eigen::numext::uint32_t;
using Eigen::numext::uint64_t;
const double pio2_62 = 3.4061215800865545e-19; // pi/2 * 2^-62
const uint64_t zero_dot_five = uint64_t(1) << 61; // 0.5 in 2.62-bit fixed-point format
// 192 bits of 2/pi for Payne-Hanek reduction
// Bits are introduced by packet of 8 to enable aligned reads.
static const uint32_t two_over_pi[] = {
0x00000028, 0x000028be, 0x0028be60, 0x28be60db, 0xbe60db93, 0x60db9391, 0xdb939105, 0x9391054a, 0x91054a7f,
0x054a7f09, 0x4a7f09d5, 0x7f09d5f4, 0x09d5f47d, 0xd5f47d4d, 0xf47d4d37, 0x7d4d3770, 0x4d377036, 0x377036d8,
0x7036d8a5, 0x36d8a566, 0xd8a5664f, 0xa5664f10, 0x664f10e4, 0x4f10e410, 0x10e41000, 0xe4100000};
uint32_t xi = numext::bit_cast<uint32_t>(xf);
// Below, -118 = -126 + 8.
// -126 is to get the exponent,
// +8 is to enable alignment of 2/pi's bits on 8 bits.
// This is possible because the fractional part of x as only 24 meaningful bits.
uint32_t e = (xi >> 23) - 118;
// Extract the mantissa and shift it to align it wrt the exponent
xi = ((xi & 0x007fffffu) | 0x00800000u) << (e & 0x7);
uint32_t i = e >> 3;
uint32_t twoopi_1 = two_over_pi[i - 1];
uint32_t twoopi_2 = two_over_pi[i + 3];
uint32_t twoopi_3 = two_over_pi[i + 7];
// Compute x * 2/pi in 2.62-bit fixed-point format.
uint64_t p;
p = uint64_t(xi) * twoopi_3;
p = uint64_t(xi) * twoopi_2 + (p >> 32);
p = (uint64_t(xi * twoopi_1) << 32) + p;
// Round to nearest: add 0.5 and extract integral part.
uint64_t q = (p + zero_dot_five) >> 62;
*quadrant = int(q);
// Now it remains to compute "r = x - q*pi/2" with high accuracy,
// since we have p=x/(pi/2) with high accuracy, we can more efficiently compute r as:
// r = (p-q)*pi/2,
// where the product can be be carried out with sufficient accuracy using double precision.
p -= q << 62;
return float(double(int64_t(p)) * pio2_62);
}
template <bool ComputeSine, typename Packet, bool ComputeBoth = false>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
#if EIGEN_COMP_GNUC_STRICT
__attribute__((optimize("-fno-unsafe-math-optimizations")))
#endif
Packet
psincos_float(const Packet& _x) {
typedef typename unpacket_traits<Packet>::integer_packet PacketI;
const Packet cst_2oPI = pset1<Packet>(0.636619746685028076171875f); // 2/PI
const Packet cst_rounding_magic = pset1<Packet>(12582912); // 2^23 for rounding
const PacketI csti_1 = pset1<PacketI>(1);
const Packet cst_sign_mask = pset1frombits<Packet>(static_cast<Eigen::numext::uint32_t>(0x80000000u));
Packet x = pabs(_x);
// Scale x by 2/Pi to find x's octant.
Packet y = pmul(x, cst_2oPI);
// Rounding trick to find nearest integer:
Packet y_round = padd(y, cst_rounding_magic);
EIGEN_OPTIMIZATION_BARRIER(y_round)
PacketI y_int = preinterpret<PacketI>(y_round); // last 23 digits represent integer (if abs(x)<2^24)
y = psub(y_round, cst_rounding_magic); // nearest integer to x * (2/pi)
// Subtract y * Pi/2 to reduce x to the interval -Pi/4 <= x <= +Pi/4
// using "Extended precision modular arithmetic"
#if defined(EIGEN_VECTORIZE_FMA)
// This version requires true FMA for high accuracy.
// It provides a max error of 1ULP up to (with absolute_error < 5.9605e-08):
const float huge_th = ComputeSine ? 117435.992f : 71476.0625f;
x = pmadd(y, pset1<Packet>(-1.57079601287841796875f), x);
x = pmadd(y, pset1<Packet>(-3.1391647326017846353352069854736328125e-07f), x);
x = pmadd(y, pset1<Packet>(-5.390302529957764765544681040410068817436695098876953125e-15f), x);
#else
// Without true FMA, the previous set of coefficients maintain 1ULP accuracy
// up to x<15.7 (for sin), but accuracy is immediately lost for x>15.7.
// We thus use one more iteration to maintain 2ULPs up to reasonably large inputs.
// The following set of coefficients maintain 1ULP up to 9.43 and 14.16 for sin and cos respectively.
// and 2 ULP up to:
const float huge_th = ComputeSine ? 25966.f : 18838.f;
x = pmadd(y, pset1<Packet>(-1.5703125), x); // = 0xbfc90000
EIGEN_OPTIMIZATION_BARRIER(x)
x = pmadd(y, pset1<Packet>(-0.000483989715576171875), x); // = 0xb9fdc000
EIGEN_OPTIMIZATION_BARRIER(x)
x = pmadd(y, pset1<Packet>(1.62865035235881805419921875e-07), x); // = 0x342ee000
x = pmadd(y, pset1<Packet>(5.5644315544167710640977020375430583953857421875e-11), x); // = 0x2e74b9ee
// For the record, the following set of coefficients maintain 2ULP up
// to a slightly larger range:
// const float huge_th = ComputeSine ? 51981.f : 39086.125f;
// but it slightly fails to maintain 1ULP for two values of sin below pi.
// x = pmadd(y, pset1<Packet>(-3.140625/2.), x);
// x = pmadd(y, pset1<Packet>(-0.00048351287841796875), x);
// x = pmadd(y, pset1<Packet>(-3.13855707645416259765625e-07), x);
// x = pmadd(y, pset1<Packet>(-6.0771006282767103812147979624569416046142578125e-11), x);
// For the record, with only 3 iterations it is possible to maintain
// 1 ULP up to 3PI (maybe more) and 2ULP up to 255.
// The coefficients are: 0xbfc90f80, 0xb7354480, 0x2e74b9ee
#endif
if (predux_any(pcmp_le(pset1<Packet>(huge_th), pabs(_x)))) {
const int PacketSize = unpacket_traits<Packet>::size;
EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) float vals[PacketSize];
EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) float x_cpy[PacketSize];
EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) Eigen::numext::int32_t y_int2[PacketSize];
pstoreu(vals, pabs(_x));
pstoreu(x_cpy, x);
pstoreu(y_int2, y_int);
for (int k = 0; k < PacketSize; ++k) {
float val = vals[k];
if (val >= huge_th && (numext::isfinite)(val)) x_cpy[k] = trig_reduce_huge(val, &y_int2[k]);
}
x = ploadu<Packet>(x_cpy);
y_int = ploadu<PacketI>(y_int2);
}
// Compute the sign to apply to the polynomial.
// sin: sign = second_bit(y_int) xor signbit(_x)
// cos: sign = second_bit(y_int+1)
Packet sign_bit = ComputeSine ? pxor(_x, preinterpret<Packet>(plogical_shift_left<30>(y_int)))
: preinterpret<Packet>(plogical_shift_left<30>(padd(y_int, csti_1)));
sign_bit = pand(sign_bit, cst_sign_mask); // clear all but left most bit
// Get the polynomial selection mask from the second bit of y_int
// We'll calculate both (sin and cos) polynomials and then select from the two.
Packet poly_mask = preinterpret<Packet>(pcmp_eq(pand(y_int, csti_1), pzero(y_int)));
Packet x2 = pmul(x, x);
// Evaluate the cos(x) polynomial. (-Pi/4 <= x <= Pi/4)
Packet y1 = pset1<Packet>(2.4372266125283204019069671630859375e-05f);
y1 = pmadd(y1, x2, pset1<Packet>(-0.00138865201734006404876708984375f));
y1 = pmadd(y1, x2, pset1<Packet>(0.041666619479656219482421875f));
y1 = pmadd(y1, x2, pset1<Packet>(-0.5f));
y1 = pmadd(y1, x2, pset1<Packet>(1.f));
// Evaluate the sin(x) polynomial. (Pi/4 <= x <= Pi/4)
// octave/matlab code to compute those coefficients:
// x = (0:0.0001:pi/4)';
// A = [x.^3 x.^5 x.^7];
// w = ((1.-(x/(pi/4)).^2).^5)*2000+1; # weights trading relative accuracy
// c = (A'*diag(w)*A)\(A'*diag(w)*(sin(x)-x)); # weighted LS, linear coeff forced to 1
// printf('%.64f\n %.64f\n%.64f\n', c(3), c(2), c(1))
//
Packet y2 = pset1<Packet>(-0.0001959234114083702898469196984621021329076029360294342041015625f);
y2 = pmadd(y2, x2, pset1<Packet>(0.0083326873655616851693794799871284340042620897293090820312500000f));
y2 = pmadd(y2, x2, pset1<Packet>(-0.1666666203982298255503735617821803316473960876464843750000000000f));
y2 = pmul(y2, x2);
y2 = pmadd(y2, x, x);
// Select the correct result from the two polynomials.
if (ComputeBoth) {
Packet peven = peven_mask(x);
Packet ysin = pselect(poly_mask, y2, y1);
Packet ycos = pselect(poly_mask, y1, y2);
Packet sign_bit_sin = pxor(_x, preinterpret<Packet>(plogical_shift_left<30>(y_int)));
Packet sign_bit_cos = preinterpret<Packet>(plogical_shift_left<30>(padd(y_int, csti_1)));
sign_bit_sin = pand(sign_bit_sin, cst_sign_mask); // clear all but left most bit
sign_bit_cos = pand(sign_bit_cos, cst_sign_mask); // clear all but left most bit
y = pselect(peven, pxor(ysin, sign_bit_sin), pxor(ycos, sign_bit_cos));
} else {
y = ComputeSine ? pselect(poly_mask, y2, y1) : pselect(poly_mask, y1, y2);
y = pxor(y, sign_bit);
}
// Update the sign and filter huge inputs
return y;
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin_float(const Packet& x) {
return psincos_float<true>(x);
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos_float(const Packet& x) {
return psincos_float<false>(x);
}
// Trigonometric argument reduction for double for inputs smaller than 15.
// Reduces trigonometric arguments for double inputs where x < 15. Given an argument x and its corresponding quadrant
// count n, the function computes and returns the reduced argument t such that x = n * pi/2 + t.
template <typename Packet>
Packet trig_reduce_small_double(const Packet& x, const Packet& q) {
// Pi/2 split into 2 values
const Packet cst_pio2_a = pset1<Packet>(-1.570796325802803);
const Packet cst_pio2_b = pset1<Packet>(-9.920935184482005e-10);
Packet t;
t = pmadd(cst_pio2_a, q, x);
t = pmadd(cst_pio2_b, q, t);
return t;
}
// Trigonometric argument reduction for double for inputs smaller than 1e14.
// Reduces trigonometric arguments for double inputs where x < 1e14. Given an argument x and its corresponding quadrant
// count n, the function computes and returns the reduced argument t such that x = n * pi/2 + t.
template <typename Packet>
Packet trig_reduce_medium_double(const Packet& x, const Packet& q_high, const Packet& q_low) {
// Pi/2 split into 4 values
const Packet cst_pio2_a = pset1<Packet>(-1.570796325802803);
const Packet cst_pio2_b = pset1<Packet>(-9.920935184482005e-10);
const Packet cst_pio2_c = pset1<Packet>(-6.123234014771656e-17);
const Packet cst_pio2_d = pset1<Packet>(1.903488962019325e-25);
Packet t;
t = pmadd(cst_pio2_a, q_high, x);
t = pmadd(cst_pio2_a, q_low, t);
t = pmadd(cst_pio2_b, q_high, t);
t = pmadd(cst_pio2_b, q_low, t);
t = pmadd(cst_pio2_c, q_high, t);
t = pmadd(cst_pio2_c, q_low, t);
t = pmadd(cst_pio2_d, padd(q_low, q_high), t);
return t;
}
template <bool ComputeSine, typename Packet, bool ComputeBoth = false>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS
#if EIGEN_COMP_GNUC_STRICT
__attribute__((optimize("-fno-unsafe-math-optimizations")))
#endif
Packet
psincos_double(const Packet& x) {
typedef typename unpacket_traits<Packet>::integer_packet PacketI;
typedef typename unpacket_traits<PacketI>::type ScalarI;
const Packet cst_sign_mask = pset1frombits<Packet>(static_cast<Eigen::numext::uint64_t>(0x8000000000000000u));
// If the argument is smaller than this value, use a simpler argument reduction
const double small_th = 15;
// If the argument is bigger than this value, use the non-vectorized std version
const double huge_th = 1e14;
const Packet cst_2oPI = pset1<Packet>(0.63661977236758134307553505349006); // 2/PI
// Integer Packet constants
const PacketI cst_one = pset1<PacketI>(ScalarI(1));
// Constant for splitting
const Packet cst_split = pset1<Packet>(1 << 24);
Packet x_abs = pabs(x);
// Scale x by 2/Pi
PacketI q_int;
Packet s;
// TODO Implement huge angle argument reduction
if (EIGEN_PREDICT_FALSE(predux_any(pcmp_le(pset1<Packet>(small_th), x_abs)))) {
Packet q_high = pmul(pfloor(pmul(x_abs, pdiv(cst_2oPI, cst_split))), cst_split);
Packet q_low_noround = psub(pmul(x_abs, cst_2oPI), q_high);
q_int = pcast<Packet, PacketI>(padd(q_low_noround, pset1<Packet>(0.5)));
Packet q_low = pcast<PacketI, Packet>(q_int);
s = trig_reduce_medium_double(x_abs, q_high, q_low);
} else {
Packet qval_noround = pmul(x_abs, cst_2oPI);
q_int = pcast<Packet, PacketI>(padd(qval_noround, pset1<Packet>(0.5)));
Packet q = pcast<PacketI, Packet>(q_int);
s = trig_reduce_small_double(x_abs, q);
}
// All the upcoming approximating polynomials have even exponents
Packet ss = pmul(s, s);
// Padé approximant of cos(x)
// Assuring < 1 ULP error on the interval [-pi/4, pi/4]
// cos(x) ~= (80737373*x^8 - 13853547000*x^6 + 727718024880*x^4 - 11275015752000*x^2 + 23594700729600)/(147173*x^8 +
// 39328920*x^6 + 5772800880*x^4 + 522334612800*x^2 + 23594700729600)
// MATLAB code to compute those coefficients:
// syms x;
// cosf = @(x) cos(x);
// pade_cosf = pade(cosf(x), x, 0, 'Order', 8)
Packet sc1_num = pmadd(ss, pset1<Packet>(80737373), pset1<Packet>(-13853547000));
Packet sc2_num = pmadd(sc1_num, ss, pset1<Packet>(727718024880));
Packet sc3_num = pmadd(sc2_num, ss, pset1<Packet>(-11275015752000));
Packet sc4_num = pmadd(sc3_num, ss, pset1<Packet>(23594700729600));
Packet sc1_denum = pmadd(ss, pset1<Packet>(147173), pset1<Packet>(39328920));
Packet sc2_denum = pmadd(sc1_denum, ss, pset1<Packet>(5772800880));
Packet sc3_denum = pmadd(sc2_denum, ss, pset1<Packet>(522334612800));
Packet sc4_denum = pmadd(sc3_denum, ss, pset1<Packet>(23594700729600));
Packet scos = pdiv(sc4_num, sc4_denum);
// Padé approximant of sin(x)
// Assuring < 1 ULP error on the interval [-pi/4, pi/4]
// sin(x) ~= (x*(4585922449*x^8 - 1066023933480*x^6 + 83284044283440*x^4 - 2303682236856000*x^2 +
// 15605159573203200))/(45*(1029037*x^8 + 345207016*x^6 + 61570292784*x^4 + 6603948711360*x^2 + 346781323848960))
// MATLAB code to compute those coefficients:
// syms x;
// sinf = @(x) sin(x);
// pade_sinf = pade(sinf(x), x, 0, 'Order', 8, 'OrderMode', 'relative')
Packet ss1_num = pmadd(ss, pset1<Packet>(4585922449), pset1<Packet>(-1066023933480));
Packet ss2_num = pmadd(ss1_num, ss, pset1<Packet>(83284044283440));
Packet ss3_num = pmadd(ss2_num, ss, pset1<Packet>(-2303682236856000));
Packet ss4_num = pmadd(ss3_num, ss, pset1<Packet>(15605159573203200));
Packet ss1_denum = pmadd(ss, pset1<Packet>(1029037), pset1<Packet>(345207016));
Packet ss2_denum = pmadd(ss1_denum, ss, pset1<Packet>(61570292784));
Packet ss3_denum = pmadd(ss2_denum, ss, pset1<Packet>(6603948711360));
Packet ss4_denum = pmadd(ss3_denum, ss, pset1<Packet>(346781323848960));
Packet ssin = pdiv(pmul(s, ss4_num), pmul(pset1<Packet>(45), ss4_denum));
Packet poly_mask = preinterpret<Packet>(pcmp_eq(pand(q_int, cst_one), pzero(q_int)));
Packet sign_sin = pxor(x, preinterpret<Packet>(plogical_shift_left<62>(q_int)));
Packet sign_cos = preinterpret<Packet>(plogical_shift_left<62>(padd(q_int, cst_one)));
Packet sign_bit, sFinalRes;
if (ComputeBoth) {
Packet peven = peven_mask(x);
sign_bit = pselect((s), sign_sin, sign_cos);
sFinalRes = pselect(pxor(peven, poly_mask), ssin, scos);
} else {
sign_bit = ComputeSine ? sign_sin : sign_cos;
sFinalRes = ComputeSine ? pselect(poly_mask, ssin, scos) : pselect(poly_mask, scos, ssin);
}
sign_bit = pand(sign_bit, cst_sign_mask); // clear all but left most bit
sFinalRes = pxor(sFinalRes, sign_bit);
// If the inputs values are higher than that a value that the argument reduction can currently address, compute them
// using std::sin and std::cos
// TODO Remove it when huge angle argument reduction is implemented
if (EIGEN_PREDICT_FALSE(predux_any(pcmp_le(pset1<Packet>(huge_th), x_abs)))) {
const int PacketSize = unpacket_traits<Packet>::size;
EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) double sincos_vals[PacketSize];
EIGEN_ALIGN_TO_BOUNDARY(sizeof(Packet)) double x_cpy[PacketSize];
pstoreu(x_cpy, x);
pstoreu(sincos_vals, sFinalRes);
for (int k = 0; k < PacketSize; ++k) {
double val = x_cpy[k];
if (std::abs(val) > huge_th && (numext::isfinite)(val)) {
if (ComputeBoth)
sincos_vals[k] = k % 2 == 0 ? std::sin(val) : std::cos(val);
else
sincos_vals[k] = ComputeSine ? std::sin(val) : std::cos(val);
}
}
sFinalRes = ploadu<Packet>(sincos_vals);
}
return sFinalRes;
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psin_double(const Packet& x) {
return psincos_double<true>(x);
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pcos_double(const Packet& x) {
return psincos_double<false>(x);
}
// Generic implementation of acos(x).
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pacos_float(const Packet& x_in) {
typedef typename unpacket_traits<Packet>::type Scalar;
static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_pi = pset1<Packet>(Scalar(EIGEN_PI));
const Packet p6 = pset1<Packet>(Scalar(2.36423197202384471893310546875e-3));
const Packet p5 = pset1<Packet>(Scalar(-1.1368644423782825469970703125e-2));
const Packet p4 = pset1<Packet>(Scalar(2.717843465507030487060546875e-2));
const Packet p3 = pset1<Packet>(Scalar(-4.8969544470310211181640625e-2));
const Packet p2 = pset1<Packet>(Scalar(8.8804088532924652099609375e-2));
const Packet p1 = pset1<Packet>(Scalar(-0.214591205120086669921875));
const Packet p0 = pset1<Packet>(Scalar(1.57079637050628662109375));
// For x in [0:1], we approximate acos(x)/sqrt(1-x), which is a smooth
// function, by a 6'th order polynomial.
// For x in [-1:0) we use that acos(-x) = pi - acos(x).
const Packet neg_mask = psignbit(x_in);
const Packet abs_x = pabs(x_in);
// Evaluate the polynomial using Horner's rule:
// P(x) = p0 + x * (p1 + x * (p2 + ... (p5 + x * p6)) ... ) .
// We evaluate even and odd terms independently to increase
// instruction level parallelism.
Packet x2 = pmul(x_in, x_in);
Packet p_even = pmadd(p6, x2, p4);
Packet p_odd = pmadd(p5, x2, p3);
p_even = pmadd(p_even, x2, p2);
p_odd = pmadd(p_odd, x2, p1);
p_even = pmadd(p_even, x2, p0);
Packet p = pmadd(p_odd, abs_x, p_even);
// The polynomial approximates acos(x)/sqrt(1-x), so
// multiply by sqrt(1-x) to get acos(x).
// Conveniently returns NaN for arguments outside [-1:1].
Packet denom = psqrt(psub(cst_one, abs_x));
Packet result = pmul(denom, p);
// Undo mapping for negative arguments.
return pselect(neg_mask, psub(cst_pi, result), result);
}
// Generic implementation of asin(x).
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pasin_float(const Packet& x_in) {
typedef typename unpacket_traits<Packet>::type Scalar;
static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
constexpr float kPiOverTwo = static_cast<float>(EIGEN_PI / 2);
const Packet cst_half = pset1<Packet>(0.5f);
const Packet cst_one = pset1<Packet>(1.0f);
const Packet cst_two = pset1<Packet>(2.0f);
const Packet cst_pi_over_two = pset1<Packet>(kPiOverTwo);
const Packet abs_x = pabs(x_in);
const Packet sign_mask = pandnot(x_in, abs_x);
const Packet invalid_mask = pcmp_lt(cst_one, abs_x);
// For arguments |x| > 0.5, we map x back to [0:0.5] using
// the transformation x_large = sqrt(0.5*(1-x)), and use the
// identity
// asin(x) = pi/2 - 2 * asin( sqrt( 0.5 * (1 - x)))
const Packet x_large = psqrt(pnmadd(cst_half, abs_x, cst_half));
const Packet large_mask = pcmp_lt(cst_half, abs_x);
const Packet x = pselect(large_mask, x_large, abs_x);
const Packet x2 = pmul(x, x);
// For |x| < 0.5 approximate asin(x)/x by an 8th order polynomial with
// even terms only.
constexpr float alpha[] = {5.08838854730129241943359375e-2f, 3.95139865577220916748046875e-2f,
7.550220191478729248046875e-2f, 0.16664917767047882080078125f, 1.00000011920928955078125f};
Packet p = ppolevl<Packet, 4>::run(x2, alpha);
p = pmul(p, x);
const Packet p_large = pnmadd(cst_two, p, cst_pi_over_two);
p = pselect(large_mask, p_large, p);
// Flip the sign for negative arguments.
p = pxor(p, sign_mask);
// Return NaN for arguments outside [-1:1].
return por(invalid_mask, p);
}
template <typename Scalar>
struct patan_reduced {
template <typename Packet>
static EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet run(const Packet& x);
};
template <>
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan_reduced<double>::run(const Packet& x) {
constexpr double alpha[] = {2.6667153866462208e-05, 3.0917513112462781e-03, 5.2574296781008604e-02,
3.0409318473444424e-01, 7.5365702534987022e-01, 8.2704055405494614e-01,
3.3004361289279920e-01};
constexpr double beta[] = {
2.7311202462436667e-04, 1.0899150928962708e-02, 1.1548932646420353e-01, 4.9716458728465573e-01, 1.0,
9.3705509168587852e-01, 3.3004361289279920e-01};
Packet x2 = pmul(x, x);
Packet p = ppolevl<Packet, 6>::run(x2, alpha);
Packet q = ppolevl<Packet, 6>::run(x2, beta);
return pmul(x, pdiv(p, q));
}
// Computes elementwise atan(x) for x in [-1:1] with 2 ulp accuracy.
template <>
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patan_reduced<float>::run(const Packet& x) {
constexpr float alpha[] = {1.12026982009410858154296875e-01f, 7.296695709228515625e-01f, 8.109951019287109375e-01f};
constexpr float beta[] = {1.00917108356952667236328125e-02f, 2.8318560123443603515625e-01f, 1.0f,
8.109951019287109375e-01f};
Packet x2 = pmul(x, x);
Packet p = ppolevl<Packet, 2>::run(x2, alpha);
Packet q = ppolevl<Packet, 3>::run(x2, beta);
return pmul(x, pdiv(p, q));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_atan(const Packet& x_in) {
typedef typename unpacket_traits<Packet>::type Scalar;
constexpr Scalar kPiOverTwo = static_cast<Scalar>(EIGEN_PI / 2);
const Packet cst_signmask = pset1<Packet>(-Scalar(0));
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_pi_over_two = pset1<Packet>(kPiOverTwo);
// "Large": For |x| > 1, use atan(1/x) = sign(x)*pi/2 - atan(x).
// "Small": For |x| <= 1, approximate atan(x) directly by a polynomial
// calculated using Rminimax.
const Packet abs_x = pabs(x_in);
const Packet x_signmask = pand(x_in, cst_signmask);
const Packet large_mask = pcmp_lt(cst_one, abs_x);
const Packet x = pselect(large_mask, preciprocal(abs_x), abs_x);
const Packet p = patan_reduced<Scalar>::run(x);
// Apply transformations according to the range reduction masks.
Packet result = pselect(large_mask, psub(cst_pi_over_two, p), p);
// Return correct sign
return pxor(result, x_signmask);
}
/** \internal \returns the hyperbolic tan of \a a (coeff-wise)
Doesn't do anything fancy, just a 9/8-degree rational interpolant which
is accurate up to a couple of ulps in the (approximate) range [-8, 8],
outside of which tanh(x) = +/-1 in single precision. The input is clamped
to the range [-c, c]. The value c is chosen as the smallest value where
the approximation evaluates to exactly 1.
This implementation works on both scalars and packets.
*/
template <typename T>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS T ptanh_float(const T& a_x) {
// Clamp the inputs to the range [-c, c] and set everything
// outside that range to 1.0. The value c is chosen as the smallest
// floating point argument such that the approximation is exactly 1.
// This saves clamping the value at the end.
#ifdef EIGEN_VECTORIZE_FMA
const T plus_clamp = pset1<T>(8.01773357391357422f);
const T minus_clamp = pset1<T>(-8.01773357391357422f);
#else
const T plus_clamp = pset1<T>(7.90738964080810547f);
const T minus_clamp = pset1<T>(-7.90738964080810547f);
#endif
const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
// The following rational approximation was generated by rminimax
// (https://gitlab.inria.fr/sfilip/rminimax) using the following
// command:
// $ ratapprox --function="tanh(x)" --dom='[-8.67,8.67]' --num="odd"
// --den="even" --type="[9,8]" --numF="[SG]" --denF="[SG]" --log
// --output=tanhf.sollya --dispCoeff="dec"
// The monomial coefficients of the numerator polynomial (odd).
constexpr float alpha[] = {1.394553628e-8f, 2.102733560e-5f, 3.520756727e-3f, 1.340216100e-1f};
// The monomial coefficients of the denominator polynomial (even).
constexpr float beta[] = {8.015776984e-7f, 3.326951409e-4f, 2.597254514e-2f, 4.673548340e-1f, 1.0f};
// Since the polynomials are odd/even, we need x^2.
const T x2 = pmul(x, x);
const T x3 = pmul(x2, x);
T p = ppolevl<T, 3>::run(x2, alpha);
T q = ppolevl<T, 4>::run(x2, beta);
// Take advantage of the fact that the constant term in p is 1 to compute
// x*(x^2*p + 1) = x^3 * p + x.
p = pmadd(x3, p, x);
// Divide the numerator by the denominator.
return pdiv(p, q);
}
/** \internal \returns the hyperbolic tan of \a a (coeff-wise)
This uses a 19/18-degree rational interpolant which
is accurate up to a couple of ulps in the (approximate) range [-18.7, 18.7],
outside of which tanh(x) = +/-1 in single precision. The input is clamped
to the range [-c, c]. The value c is chosen as the smallest value where
the approximation evaluates to exactly 1.
This implementation works on both scalars and packets.
*/
template <typename T>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS T ptanh_double(const T& a_x) {
// Clamp the inputs to the range [-c, c] and set everything
// outside that range to 1.0. The value c is chosen as the smallest
// floating point argument such that the approximation is exactly 1.
// This saves clamping the value at the end.
#ifdef EIGEN_VECTORIZE_FMA
const T plus_clamp = pset1<T>(17.6610191624600077);
const T minus_clamp = pset1<T>(-17.6610191624600077);
#else
const T plus_clamp = pset1<T>(17.714196154005176);
const T minus_clamp = pset1<T>(-17.714196154005176);
#endif
const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
// The following rational approximation was generated by rminimax
// (https://gitlab.inria.fr/sfilip/rminimax) using the following
// command:
// $ ./ratapprox --function="tanh(x)" --dom='[-18.72,18.72]'
// --num="odd" --den="even" --type="[19,18]" --numF="[D]"
// --denF="[D]" --log --output=tanh.sollya --dispCoeff="dec"
// The monomial coefficients of the numerator polynomial (odd).
constexpr double alpha[] = {2.6158007860482230e-23, 7.6534862268749319e-19, 3.1309488231386680e-15,
4.2303918148209176e-12, 2.4618379131293676e-09, 6.8644367682497074e-07,
9.3839087674268880e-05, 5.9809711724441161e-03, 1.5184719640284322e-01};
// The monomial coefficients of the denominator polynomial (even).
constexpr double beta[] = {6.463747022670968018e-21, 5.782506856739003571e-17,
1.293019623712687916e-13, 1.123643448069621992e-10,
4.492975677839633985e-08, 8.785185266237658698e-06,
8.295161192716231542e-04, 3.437448108450402717e-02,
4.851805297361760360e-01, 1.0};
// Since the polynomials are odd/even, we need x^2.
const T x2 = pmul(x, x);
const T x3 = pmul(x2, x);
// Interleave the evaluation of the numerator polynomial p and
// denominator polynomial q.
T p = ppolevl<T, 8>::run(x2, alpha);
T q = ppolevl<T, 9>::run(x2, beta);
// Take advantage of the fact that the constant term in p is 1 to compute
// x*(x^2*p + 1) = x^3 * p + x.
p = pmadd(x3, p, x);
// Divide the numerator by the denominator.
return pdiv(p, q);
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patanh_float(const Packet& x) {
typedef typename unpacket_traits<Packet>::type Scalar;
static_assert(std::is_same<Scalar, float>::value, "Scalar type must be float");
// For |x| in [0:0.5] we use a polynomial approximation of the form
// P(x) = x + x^3*(alpha[4] + x^2 * (alpha[3] + x^2 * (... x^2 * alpha[0]) ... )).
constexpr float alpha[] = {0.1819281280040740966796875f, 8.2311116158962249755859375e-2f,
0.14672131836414337158203125f, 0.1997792422771453857421875f, 0.3333373963832855224609375f};
const Packet x2 = pmul(x, x);
const Packet x3 = pmul(x, x2);
Packet p = ppolevl<Packet, 4>::run(x2, alpha);
p = pmadd(x3, p, x);
// For |x| in ]0.5:1.0] we use atanh = 0.5*ln((1+x)/(1-x));
const Packet half = pset1<Packet>(0.5f);
const Packet one = pset1<Packet>(1.0f);
Packet r = pdiv(padd(one, x), psub(one, x));
r = pmul(half, plog(r));
const Packet x_gt_half = pcmp_le(half, pabs(x));
const Packet x_eq_one = pcmp_eq(one, pabs(x));
const Packet x_gt_one = pcmp_lt(one, pabs(x));
const Packet sign_mask = pset1<Packet>(-0.0f);
const Packet x_sign = pand(sign_mask, x);
const Packet inf = pset1<Packet>(std::numeric_limits<float>::infinity());
return por(x_gt_one, pselect(x_eq_one, por(x_sign, inf), pselect(x_gt_half, r, p)));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet patanh_double(const Packet& x) {
typedef typename unpacket_traits<Packet>::type Scalar;
static_assert(std::is_same<Scalar, double>::value, "Scalar type must be double");
// For x in [-0.5:0.5] we use a rational approximation of the form
// R(x) = x + x^3*P(x^2)/Q(x^2), where P is or order 4 and Q is of order 5.
constexpr double alpha[] = {3.3071338469301391e-03, -4.7129526768798737e-02, 1.8185306179826699e-01,
-2.5949536095445679e-01, 1.2306328729812676e-01};
constexpr double beta[] = {-3.8679974580640881e-03, 7.6391885763341910e-02, -4.2828141436397615e-01,
9.8733495886883648e-01, -1.0000000000000000e+00, 3.6918986189438030e-01};
const Packet x2 = pmul(x, x);
const Packet x3 = pmul(x, x2);
Packet p = ppolevl<Packet, 4>::run(x2, alpha);
Packet q = ppolevl<Packet, 5>::run(x2, beta);
Packet y_small = pmadd(x3, pdiv(p, q), x);
// For |x| in ]0.5:1.0] we use atanh = 0.5*ln((1+x)/(1-x));
const Packet half = pset1<Packet>(0.5);
const Packet one = pset1<Packet>(1.0);
Packet y_large = pdiv(padd(one, x), psub(one, x));
y_large = pmul(half, plog(y_large));
const Packet x_gt_half = pcmp_le(half, pabs(x));
const Packet x_eq_one = pcmp_eq(one, pabs(x));
const Packet x_gt_one = pcmp_lt(one, pabs(x));
const Packet sign_mask = pset1<Packet>(-0.0);
const Packet x_sign = pand(sign_mask, x);
const Packet inf = pset1<Packet>(std::numeric_limits<double>::infinity());
return por(x_gt_one, pselect(x_eq_one, por(x_sign, inf), pselect(x_gt_half, y_large, y_small)));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pdiv_complex(const Packet& x, const Packet& y) {
typedef typename unpacket_traits<Packet>::as_real RealPacket;
// In the following we annotate the code for the case where the inputs
// are a pair length-2 SIMD vectors representing a single pair of complex
// numbers x = a + i*b, y = c + i*d.
const RealPacket y_abs = pabs(y.v); // |c|, |d|
const RealPacket y_abs_flip = pcplxflip(Packet(y_abs)).v; // |d|, |c|
const RealPacket y_max = pmax(y_abs, y_abs_flip); // max(|c|, |d|), max(|c|, |d|)
const RealPacket y_scaled = pdiv(y.v, y_max); // c / max(|c|, |d|), d / max(|c|, |d|)
// Compute scaled denominator.
const RealPacket y_scaled_sq = pmul(y_scaled, y_scaled); // c'**2, d'**2
const RealPacket denom = padd(y_scaled_sq, pcplxflip(Packet(y_scaled_sq)).v);
Packet result_scaled = pmul(x, pconj(Packet(y_scaled))); // a * c' + b * d', -a * d + b * c
// Divide elementwise by denom.
result_scaled = Packet(pdiv(result_scaled.v, denom));
// Rescale result
return Packet(pdiv(result_scaled.v, y_max));
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet plog_complex(const Packet& x) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename Scalar::value_type RealScalar;
typedef typename unpacket_traits<Packet>::as_real RealPacket;
RealPacket real_mask_rp = peven_mask(x.v);
Packet real_mask(real_mask_rp);
// Real part
RealPacket x_flip = pcplxflip(x).v; // b, a
Packet x_norm = phypot_complex(x); // sqrt(a^2 + b^2), sqrt(a^2 + b^2)
RealPacket xlogr = plog(x_norm.v); // log(sqrt(a^2 + b^2)), log(sqrt(a^2 + b^2))
// Imag part
RealPacket ximg = patan2(x.v, x_flip); // atan2(a, b), atan2(b, a)
const RealPacket cst_pos_inf = pset1<RealPacket>(NumTraits<RealScalar>::infinity());
RealPacket x_abs = pabs(x.v);
RealPacket is_x_pos_inf = pcmp_eq(x_abs, cst_pos_inf);
RealPacket is_y_pos_inf = pcplxflip(Packet(is_x_pos_inf)).v;
RealPacket is_any_inf = por(is_x_pos_inf, is_y_pos_inf);
RealPacket xreal = pselect(is_any_inf, cst_pos_inf, xlogr);
Packet xres = pselect(real_mask, Packet(xreal), Packet(ximg)); // log(sqrt(a^2 + b^2)), atan2(b, a)
return xres;
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet pexp_complex(const Packet& a) {
typedef typename unpacket_traits<Packet>::as_real RealPacket;
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename Scalar::value_type RealScalar;
const RealPacket even_mask = peven_mask(a.v);
const RealPacket odd_mask = pcplxflip(Packet(even_mask)).v;
// Let a = x + iy.
// exp(a) = exp(x) * cis(y), plus some special edge-case handling.
// exp(x):
RealPacket x = pand(a.v, even_mask);
x = por(x, pcplxflip(Packet(x)).v);
RealPacket expx = pexp(x); // exp(x);
// cis(y):
RealPacket y = pand(odd_mask, a.v);
y = por(y, pcplxflip(Packet(y)).v);
RealPacket cisy = psincos_float<false, RealPacket, true>(y);
cisy = pcplxflip(Packet(cisy)).v; // cos(y) + i * sin(y)
const RealPacket cst_pos_inf = pset1<RealPacket>(NumTraits<RealScalar>::infinity());
const RealPacket cst_neg_inf = pset1<RealPacket>(-NumTraits<RealScalar>::infinity());
// If x is -inf, we know that cossin(y) is bounded,
// so the result is (0, +/-0), where the sign of the imaginary part comes
// from the sign of cossin(y).
RealPacket cisy_sign = por(pandnot(cisy, pabs(cisy)), pset1<RealPacket>(RealScalar(1)));
cisy = pselect(pcmp_eq(x, cst_neg_inf), cisy_sign, cisy);
// If x is inf, and cos(y) has unknown sign (y is inf or NaN), the result
// is (+/-inf, NaN), where the signs are undetermined (take the sign of y).
RealPacket y_sign = por(pandnot(y, pabs(y)), pset1<RealPacket>(RealScalar(1)));
cisy = pselect(pand(pcmp_eq(x, cst_pos_inf), pisnan(cisy)), pand(y_sign, even_mask), cisy);
Packet result = Packet(pmul(expx, cisy));
// If y is +/- 0, the input is real, so take the real result for consistency.
result = pselect(Packet(pcmp_eq(y, pzero(y))), Packet(por(pand(expx, even_mask), pand(y, odd_mask))), result);
return result;
}
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet psqrt_complex(const Packet& a) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename Scalar::value_type RealScalar;
typedef typename unpacket_traits<Packet>::as_real RealPacket;
// Computes the principal sqrt of the complex numbers in the input.
//
// For example, for packets containing 2 complex numbers stored in interleaved format
// a = [a0, a1] = [x0, y0, x1, y1],
// where x0 = real(a0), y0 = imag(a0) etc., this function returns
// b = [b0, b1] = [u0, v0, u1, v1],
// such that b0^2 = a0, b1^2 = a1.
//
// To derive the formula for the complex square roots, let's consider the equation for
// a single complex square root of the number x + i*y. We want to find real numbers
// u and v such that
// (u + i*v)^2 = x + i*y <=>
// u^2 - v^2 + i*2*u*v = x + i*v.
// By equating the real and imaginary parts we get:
// u^2 - v^2 = x
// 2*u*v = y.
//
// For x >= 0, this has the numerically stable solution
// u = sqrt(0.5 * (x + sqrt(x^2 + y^2)))
// v = 0.5 * (y / u)
// and for x < 0,
// v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2)))
// u = 0.5 * (y / v)
//
// To avoid unnecessary over- and underflow, we compute sqrt(x^2 + y^2) as
// l = max(|x|, |y|) * sqrt(1 + (min(|x|, |y|) / max(|x|, |y|))^2) ,
// In the following, without lack of generality, we have annotated the code, assuming
// that the input is a packet of 2 complex numbers.
//
// Step 1. Compute l = [l0, l0, l1, l1], where
// l0 = sqrt(x0^2 + y0^2), l1 = sqrt(x1^2 + y1^2)
// To avoid over- and underflow, we use the stable formula for each hypotenuse
// l0 = (min0 == 0 ? max0 : max0 * sqrt(1 + (min0/max0)**2)),
// where max0 = max(|x0|, |y0|), min0 = min(|x0|, |y0|), and similarly for l1.
RealPacket a_abs = pabs(a.v); // [|x0|, |y0|, |x1|, |y1|]
RealPacket a_abs_flip = pcplxflip(Packet(a_abs)).v; // [|y0|, |x0|, |y1|, |x1|]
RealPacket a_max = pmax(a_abs, a_abs_flip);
RealPacket a_min = pmin(a_abs, a_abs_flip);
RealPacket a_min_zero_mask = pcmp_eq(a_min, pzero(a_min));
RealPacket a_max_zero_mask = pcmp_eq(a_max, pzero(a_max));
RealPacket r = pdiv(a_min, a_max);
const RealPacket cst_one = pset1<RealPacket>(RealScalar(1));
RealPacket l = pmul(a_max, psqrt(padd(cst_one, pmul(r, r)))); // [l0, l0, l1, l1]
// Set l to a_max if a_min is zero.
l = pselect(a_min_zero_mask, a_max, l);
// Step 2. Compute [rho0, *, rho1, *], where
// rho0 = sqrt(0.5 * (l0 + |x0|)), rho1 = sqrt(0.5 * (l1 + |x1|))
// We don't care about the imaginary parts computed here. They will be overwritten later.
const RealPacket cst_half = pset1<RealPacket>(RealScalar(0.5));
Packet rho;
rho.v = psqrt(pmul(cst_half, padd(a_abs, l)));
// Step 3. Compute [rho0, eta0, rho1, eta1], where
// eta0 = (y0 / l0) / 2, and eta1 = (y1 / l1) / 2.
// set eta = 0 of input is 0 + i0.
RealPacket eta = pandnot(pmul(cst_half, pdiv(a.v, pcplxflip(rho).v)), a_max_zero_mask);
RealPacket real_mask = peven_mask(a.v);
Packet positive_real_result;
// Compute result for inputs with positive real part.
positive_real_result.v = pselect(real_mask, rho.v, eta);
// Step 4. Compute solution for inputs with negative real part:
// [|eta0|, sign(y0)*rho0, |eta1|, sign(y1)*rho1]
const RealPacket cst_imag_sign_mask = pset1<Packet>(Scalar(RealScalar(0.0), RealScalar(-0.0))).v;
RealPacket imag_signs = pand(a.v, cst_imag_sign_mask);
Packet negative_real_result;
// Notice that rho is positive, so taking it's absolute value is a noop.
negative_real_result.v = por(pabs(pcplxflip(positive_real_result).v), imag_signs);
// Step 5. Select solution branch based on the sign of the real parts.
Packet negative_real_mask;
negative_real_mask.v = pcmp_lt(pand(real_mask, a.v), pzero(a.v));
negative_real_mask.v = por(negative_real_mask.v, pcplxflip(negative_real_mask).v);
Packet result = pselect(negative_real_mask, negative_real_result, positive_real_result);
// Step 6. Handle special cases for infinities:
// * If z is (x,+∞), the result is (+∞,+∞) even if x is NaN
// * If z is (x,-∞), the result is (+∞,-∞) even if x is NaN
// * If z is (-∞,y), the result is (0*|y|,+∞) for finite or NaN y
// * If z is (+∞,y), the result is (+∞,0*|y|) for finite or NaN y
const RealPacket cst_pos_inf = pset1<RealPacket>(NumTraits<RealScalar>::infinity());
Packet is_inf;
is_inf.v = pcmp_eq(a_abs, cst_pos_inf);
Packet is_real_inf;
is_real_inf.v = pand(is_inf.v, real_mask);
is_real_inf = por(is_real_inf, pcplxflip(is_real_inf));
// prepare packet of (+∞,0*|y|) or (0*|y|,+∞), depending on the sign of the infinite real part.
Packet real_inf_result;
real_inf_result.v = pmul(a_abs, pset1<Packet>(Scalar(RealScalar(1.0), RealScalar(0.0))).v);
real_inf_result.v = pselect(negative_real_mask.v, pcplxflip(real_inf_result).v, real_inf_result.v);
// prepare packet of (+∞,+∞) or (+∞,-∞), depending on the sign of the infinite imaginary part.
Packet is_imag_inf;
is_imag_inf.v = pandnot(is_inf.v, real_mask);
is_imag_inf = por(is_imag_inf, pcplxflip(is_imag_inf));
Packet imag_inf_result;
imag_inf_result.v = por(pand(cst_pos_inf, real_mask), pandnot(a.v, real_mask));
// unless otherwise specified, if either the real or imaginary component is nan, the entire result is nan
Packet result_is_nan = pisnan(result);
result = por(result_is_nan, result);
return pselect(is_imag_inf, imag_inf_result, pselect(is_real_inf, real_inf_result, result));
}
// \internal \returns the norm of a complex number z = x + i*y, defined as sqrt(x^2 + y^2).
// Implemented using the hypot(a,b) algorithm from https://doi.org/10.48550/arXiv.1904.09481
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet phypot_complex(const Packet& a) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename Scalar::value_type RealScalar;
typedef typename unpacket_traits<Packet>::as_real RealPacket;
const RealPacket cst_zero_rp = pset1<RealPacket>(static_cast<RealScalar>(0.0));
const RealPacket cst_minus_one_rp = pset1<RealPacket>(static_cast<RealScalar>(-1.0));
const RealPacket cst_two_rp = pset1<RealPacket>(static_cast<RealScalar>(2.0));
const RealPacket evenmask = peven_mask(a.v);
RealPacket a_abs = pabs(a.v);
RealPacket a_flip = pcplxflip(Packet(a_abs)).v; // |b|, |a|
RealPacket a_all = pselect(evenmask, a_abs, a_flip); // |a|, |a|
RealPacket b_all = pselect(evenmask, a_flip, a_abs); // |b|, |b|
RealPacket a2 = pmul(a.v, a.v); // |a^2, b^2|
RealPacket a2_flip = pcplxflip(Packet(a2)).v; // |b^2, a^2|
RealPacket h = psqrt(padd(a2, a2_flip)); // |sqrt(a^2 + b^2), sqrt(a^2 + b^2)|
RealPacket h_sq = pmul(h, h); // |a^2 + b^2, a^2 + b^2|
RealPacket a_sq = pselect(evenmask, a2, a2_flip); // |a^2, a^2|
RealPacket m_h_sq = pmul(h_sq, cst_minus_one_rp);
RealPacket m_a_sq = pmul(a_sq, cst_minus_one_rp);
RealPacket x = psub(psub(pmadd(h, h, m_h_sq), pmadd(b_all, b_all, psub(a_sq, h_sq))), pmadd(a_all, a_all, m_a_sq));
h = psub(h, pdiv(x, pmul(cst_two_rp, h))); // |h - x/(2*h), h - x/(2*h)|
// handle zero-case
RealPacket iszero = pcmp_eq(por(a_abs, a_flip), cst_zero_rp);
h = pandnot(h, iszero); // |sqrt(a^2+b^2), sqrt(a^2+b^2)|
return Packet(h); // |sqrt(a^2+b^2), sqrt(a^2+b^2)|
}
template <typename Packet>
struct psign_impl<Packet, std::enable_if_t<!NumTraits<typename unpacket_traits<Packet>::type>::IsComplex &&
!NumTraits<typename unpacket_traits<Packet>::type>::IsInteger>> {
static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_zero = pzero(a);
const Packet abs_a = pabs(a);
const Packet sign_mask = pandnot(a, abs_a);
const Packet nonzero_mask = pcmp_lt(cst_zero, abs_a);
return pselect(nonzero_mask, por(sign_mask, cst_one), abs_a);
}
};
template <typename Packet>
struct psign_impl<Packet, std::enable_if_t<!NumTraits<typename unpacket_traits<Packet>::type>::IsComplex &&
NumTraits<typename unpacket_traits<Packet>::type>::IsSigned &&
NumTraits<typename unpacket_traits<Packet>::type>::IsInteger>> {
static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_minus_one = pset1<Packet>(Scalar(-1));
const Packet cst_zero = pzero(a);
const Packet positive_mask = pcmp_lt(cst_zero, a);
const Packet positive = pand(positive_mask, cst_one);
const Packet negative_mask = pcmp_lt(a, cst_zero);
const Packet negative = pand(negative_mask, cst_minus_one);
return por(positive, negative);
}
};
template <typename Packet>
struct psign_impl<Packet, std::enable_if_t<!NumTraits<typename unpacket_traits<Packet>::type>::IsComplex &&
!NumTraits<typename unpacket_traits<Packet>::type>::IsSigned &&
NumTraits<typename unpacket_traits<Packet>::type>::IsInteger>> {
static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_zero = pzero(a);
const Packet zero_mask = pcmp_eq(cst_zero, a);
return pandnot(cst_one, zero_mask);
}
};
// \internal \returns the the sign of a complex number z, defined as z / abs(z).
template <typename Packet>
struct psign_impl<Packet, std::enable_if_t<NumTraits<typename unpacket_traits<Packet>::type>::IsComplex &&
unpacket_traits<Packet>::vectorizable>> {
static EIGEN_DEVICE_FUNC inline Packet run(const Packet& a) {
typedef typename unpacket_traits<Packet>::type Scalar;
typedef typename Scalar::value_type RealScalar;
typedef typename unpacket_traits<Packet>::as_real RealPacket;
// Step 1. Compute (for each element z = x + i*y in a)
// l = abs(z) = sqrt(x^2 + y^2).
// To avoid over- and underflow, we use the stable formula for each hypotenuse
// l = (zmin == 0 ? zmax : zmax * sqrt(1 + (zmin/zmax)**2)),
// where zmax = max(|x|, |y|), zmin = min(|x|, |y|),
RealPacket a_abs = pabs(a.v);
RealPacket a_abs_flip = pcplxflip(Packet(a_abs)).v;
RealPacket a_max = pmax(a_abs, a_abs_flip);
RealPacket a_min = pmin(a_abs, a_abs_flip);
RealPacket a_min_zero_mask = pcmp_eq(a_min, pzero(a_min));
RealPacket a_max_zero_mask = pcmp_eq(a_max, pzero(a_max));
RealPacket r = pdiv(a_min, a_max);
const RealPacket cst_one = pset1<RealPacket>(RealScalar(1));
RealPacket l = pmul(a_max, psqrt(padd(cst_one, pmul(r, r)))); // [l0, l0, l1, l1]
// Set l to a_max if a_min is zero, since the roundtrip sqrt(a_max^2) may be
// lossy.
l = pselect(a_min_zero_mask, a_max, l);
// Step 2 compute a / abs(a).
RealPacket sign_as_real = pandnot(pdiv(a.v, l), a_max_zero_mask);
Packet sign;
sign.v = sign_as_real;
return sign;
}
};
// TODO(rmlarsen): The following set of utilities for double word arithmetic
// should perhaps be refactored as a separate file, since it would be generally
// useful for special function implementation etc. Writing the algorithms in
// terms if a double word type would also make the code more readable.
// This function splits x into the nearest integer n and fractional part r,
// such that x = n + r holds exactly.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void absolute_split(const Packet& x, Packet& n, Packet& r) {
n = pround(x);
r = psub(x, n);
}
// This function computes the sum {s, r}, such that x + y = s_hi + s_lo
// holds exactly, and s_hi = fl(x+y), if |x| >= |y|.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void fast_twosum(const Packet& x, const Packet& y, Packet& s_hi, Packet& s_lo) {
s_hi = padd(x, y);
const Packet t = psub(s_hi, x);
s_lo = psub(y, t);
}
#ifdef EIGEN_VECTORIZE_FMA
// This function implements the extended precision product of
// a pair of floating point numbers. Given {x, y}, it computes the pair
// {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and
// p_hi = fl(x * y).
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void twoprod(const Packet& x, const Packet& y, Packet& p_hi, Packet& p_lo) {
p_hi = pmul(x, y);
p_lo = pmsub(x, y, p_hi);
}
// A version of twoprod that takes x, y, and fl(x*y) as input and returns the p_lo such that
// x * y = xy + p_lo holds exactly.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet twoprod_low(const Packet& x, const Packet& y, const Packet& xy) {
return pmsub(x, y, xy);
}
#else
// This function implements the Veltkamp splitting. Given a floating point
// number x it returns the pair {x_hi, x_lo} such that x_hi + x_lo = x holds
// exactly and that half of the significant of x fits in x_hi.
// This is Algorithm 3 from Jean-Michel Muller, "Elementary Functions",
// 3rd edition, Birkh\"auser, 2016.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void veltkamp_splitting(const Packet& x, Packet& x_hi, Packet& x_lo) {
typedef typename unpacket_traits<Packet>::type Scalar;
EIGEN_CONSTEXPR int shift = (NumTraits<Scalar>::digits() + 1) / 2;
const Scalar shift_scale = Scalar(uint64_t(1) << shift); // Scalar constructor not necessarily constexpr.
const Packet gamma = pmul(pset1<Packet>(shift_scale + Scalar(1)), x);
Packet rho = psub(x, gamma);
x_hi = padd(rho, gamma);
x_lo = psub(x, x_hi);
}
// This function implements Dekker's algorithm for products x * y.
// Given floating point numbers {x, y} computes the pair
// {p_hi, p_lo} such that x * y = p_hi + p_lo holds exactly and
// p_hi = fl(x * y).
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void twoprod(const Packet& x, const Packet& y, Packet& p_hi, Packet& p_lo) {
Packet x_hi, x_lo, y_hi, y_lo;
veltkamp_splitting(x, x_hi, x_lo);
veltkamp_splitting(y, y_hi, y_lo);
p_hi = pmul(x, y);
p_lo = pmadd(x_hi, y_hi, pnegate(p_hi));
p_lo = pmadd(x_hi, y_lo, p_lo);
p_lo = pmadd(x_lo, y_hi, p_lo);
p_lo = pmadd(x_lo, y_lo, p_lo);
}
// A version of twoprod that takes x, y, and fl(x*y) as input and returns the p_lo such that
// x * y = xy + p_lo holds exactly.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet twoprod_low(const Packet& x, const Packet& y, const Packet& xy) {
Packet x_hi, x_lo, y_hi, y_lo;
veltkamp_splitting(x, x_hi, x_lo);
veltkamp_splitting(y, y_hi, y_lo);
Packet p_lo = pmadd(x_hi, y_hi, pnegate(xy));
p_lo = pmadd(x_hi, y_lo, p_lo);
p_lo = pmadd(x_lo, y_hi, p_lo);
p_lo = pmadd(x_lo, y_lo, p_lo);
return p_lo;
}
#endif // EIGEN_VECTORIZE_FMA
// This function implements Dekker's algorithm for the addition
// of two double word numbers represented by {x_hi, x_lo} and {y_hi, y_lo}.
// It returns the result as a pair {s_hi, s_lo} such that
// x_hi + x_lo + y_hi + y_lo = s_hi + s_lo holds exactly.
// This is Algorithm 5 from Jean-Michel Muller, "Elementary Functions",
// 3rd edition, Birkh\"auser, 2016.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void twosum(const Packet& x_hi, const Packet& x_lo, const Packet& y_hi,
const Packet& y_lo, Packet& s_hi, Packet& s_lo) {
const Packet x_greater_mask = pcmp_lt(pabs(y_hi), pabs(x_hi));
Packet r_hi_1, r_lo_1;
fast_twosum(x_hi, y_hi, r_hi_1, r_lo_1);
Packet r_hi_2, r_lo_2;
fast_twosum(y_hi, x_hi, r_hi_2, r_lo_2);
const Packet r_hi = pselect(x_greater_mask, r_hi_1, r_hi_2);
const Packet s1 = padd(padd(y_lo, r_lo_1), x_lo);
const Packet s2 = padd(padd(x_lo, r_lo_2), y_lo);
const Packet s = pselect(x_greater_mask, s1, s2);
fast_twosum(r_hi, s, s_hi, s_lo);
}
// This is a version of twosum for double word numbers,
// which assumes that |x_hi| >= |y_hi|.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void fast_twosum(const Packet& x_hi, const Packet& x_lo, const Packet& y_hi,
const Packet& y_lo, Packet& s_hi, Packet& s_lo) {
Packet r_hi, r_lo;
fast_twosum(x_hi, y_hi, r_hi, r_lo);
const Packet s = padd(padd(y_lo, r_lo), x_lo);
fast_twosum(r_hi, s, s_hi, s_lo);
}
// This is a version of twosum for adding a floating point number x to
// double word number {y_hi, y_lo} number, with the assumption
// that |x| >= |y_hi|.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void fast_twosum(const Packet& x, const Packet& y_hi, const Packet& y_lo,
Packet& s_hi, Packet& s_lo) {
Packet r_hi, r_lo;
fast_twosum(x, y_hi, r_hi, r_lo);
const Packet s = padd(y_lo, r_lo);
fast_twosum(r_hi, s, s_hi, s_lo);
}
// This function implements the multiplication of a double word
// number represented by {x_hi, x_lo} by a floating point number y.
// It returns the result as a pair {p_hi, p_lo} such that
// (x_hi + x_lo) * y = p_hi + p_lo hold with a relative error
// of less than 2*2^{-2p}, where p is the number of significand bit
// in the floating point type.
// This is Algorithm 7 from Jean-Michel Muller, "Elementary Functions",
// 3rd edition, Birkh\"auser, 2016.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void twoprod(const Packet& x_hi, const Packet& x_lo, const Packet& y,
Packet& p_hi, Packet& p_lo) {
Packet c_hi, c_lo1;
twoprod(x_hi, y, c_hi, c_lo1);
const Packet c_lo2 = pmul(x_lo, y);
Packet t_hi, t_lo1;
fast_twosum(c_hi, c_lo2, t_hi, t_lo1);
const Packet t_lo2 = padd(t_lo1, c_lo1);
fast_twosum(t_hi, t_lo2, p_hi, p_lo);
}
// This function implements the multiplication of two double word
// numbers represented by {x_hi, x_lo} and {y_hi, y_lo}.
// It returns the result as a pair {p_hi, p_lo} such that
// (x_hi + x_lo) * (y_hi + y_lo) = p_hi + p_lo holds with a relative error
// of less than 2*2^{-2p}, where p is the number of significand bit
// in the floating point type.
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void twoprod(const Packet& x_hi, const Packet& x_lo, const Packet& y_hi,
const Packet& y_lo, Packet& p_hi, Packet& p_lo) {
Packet p_hi_hi, p_hi_lo;
twoprod(x_hi, x_lo, y_hi, p_hi_hi, p_hi_lo);
Packet p_lo_hi, p_lo_lo;
twoprod(x_hi, x_lo, y_lo, p_lo_hi, p_lo_lo);
fast_twosum(p_hi_hi, p_hi_lo, p_lo_hi, p_lo_lo, p_hi, p_lo);
}
// This function implements the division of double word {x_hi, x_lo}
// by float y. This is Algorithm 15 from "Tight and rigorous error bounds
// for basic building blocks of double-word arithmetic", Joldes, Muller, & Popescu,
// 2017. https://hal.archives-ouvertes.fr/hal-01351529
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void doubleword_div_fp(const Packet& x_hi, const Packet& x_lo, const Packet& y,
Packet& z_hi, Packet& z_lo) {
const Packet t_hi = pdiv(x_hi, y);
Packet pi_hi, pi_lo;
twoprod(t_hi, y, pi_hi, pi_lo);
const Packet delta_hi = psub(x_hi, pi_hi);
const Packet delta_t = psub(delta_hi, pi_lo);
const Packet delta = padd(delta_t, x_lo);
const Packet t_lo = pdiv(delta, y);
fast_twosum(t_hi, t_lo, z_hi, z_lo);
}
// This function computes log2(x) and returns the result as a double word.
template <typename Scalar>
struct accurate_log2 {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void operator()(const Packet& x, Packet& log2_x_hi, Packet& log2_x_lo) {
log2_x_hi = plog2(x);
log2_x_lo = pzero(x);
}
};
// This specialization uses a more accurate algorithm to compute log2(x) for
// floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~6.42e-10.
// This additional accuracy is needed to counter the error-magnification
// inherent in multiplying by a potentially large exponent in pow(x,y).
// The minimax polynomial used was calculated using the Sollya tool.
// See sollya.org.
template <>
struct accurate_log2<float> {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void operator()(const Packet& z, Packet& log2_x_hi, Packet& log2_x_lo) {
// The function log(1+x)/x is approximated in the interval
// [1/sqrt(2)-1;sqrt(2)-1] by a degree 10 polynomial of the form
// Q(x) = (C0 + x * (C1 + x * (C2 + x * (C3 + x * P(x))))),
// where the degree 6 polynomial P(x) is evaluated in single precision,
// while the remaining 4 terms of Q(x), as well as the final multiplication by x
// to reconstruct log(1+x) are evaluated in extra precision using
// double word arithmetic. C0 through C3 are extra precise constants
// stored as double words.
//
// The polynomial coefficients were calculated using Sollya commands:
// > n = 10;
// > f = log2(1+x)/x;
// > interval = [sqrt(0.5)-1;sqrt(2)-1];
// > p = fpminimax(f,n,[|double,double,double,double,single...|],interval,relative,floating);
const Packet p6 = pset1<Packet>(9.703654795885e-2f);
const Packet p5 = pset1<Packet>(-0.1690667718648f);
const Packet p4 = pset1<Packet>(0.1720575392246f);
const Packet p3 = pset1<Packet>(-0.1789081543684f);
const Packet p2 = pset1<Packet>(0.2050433009862f);
const Packet p1 = pset1<Packet>(-0.2404672354459f);
const Packet p0 = pset1<Packet>(0.2885761857032f);
const Packet C3_hi = pset1<Packet>(-0.360674142838f);
const Packet C3_lo = pset1<Packet>(-6.13283912543e-09f);
const Packet C2_hi = pset1<Packet>(0.480897903442f);
const Packet C2_lo = pset1<Packet>(-1.44861207474e-08f);
const Packet C1_hi = pset1<Packet>(-0.721347510815f);
const Packet C1_lo = pset1<Packet>(-4.84483164698e-09f);
const Packet C0_hi = pset1<Packet>(1.44269502163f);
const Packet C0_lo = pset1<Packet>(2.01711713999e-08f);
const Packet one = pset1<Packet>(1.0f);
const Packet x = psub(z, one);
// Evaluate P(x) in working precision.
// We evaluate it in multiple parts to improve instruction level
// parallelism.
Packet x2 = pmul(x, x);
Packet p_even = pmadd(p6, x2, p4);
p_even = pmadd(p_even, x2, p2);
p_even = pmadd(p_even, x2, p0);
Packet p_odd = pmadd(p5, x2, p3);
p_odd = pmadd(p_odd, x2, p1);
Packet p = pmadd(p_odd, x, p_even);
// Now evaluate the low-order tems of Q(x) in double word precision.
// In the following, due to the alternating signs and the fact that
// |x| < sqrt(2)-1, we can assume that |C*_hi| >= q_i, and use
// fast_twosum instead of the slower twosum.
Packet q_hi, q_lo;
Packet t_hi, t_lo;
// C3 + x * p(x)
twoprod(p, x, t_hi, t_lo);
fast_twosum(C3_hi, C3_lo, t_hi, t_lo, q_hi, q_lo);
// C2 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C2_hi, C2_lo, t_hi, t_lo, q_hi, q_lo);
// C1 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C1_hi, C1_lo, t_hi, t_lo, q_hi, q_lo);
// C0 + x * p(x)
twoprod(q_hi, q_lo, x, t_hi, t_lo);
fast_twosum(C0_hi, C0_lo, t_hi, t_lo, q_hi, q_lo);
// log(z) ~= x * Q(x)
twoprod(q_hi, q_lo, x, log2_x_hi, log2_x_lo);
}
};
// This specialization uses a more accurate algorithm to compute log2(x) for
// floats in [1/sqrt(2);sqrt(2)] with a relative accuracy of ~1.27e-18.
// This additional accuracy is needed to counter the error-magnification
// inherent in multiplying by a potentially large exponent in pow(x,y).
// The minimax polynomial used was calculated using the Sollya tool.
// See sollya.org.
template <>
struct accurate_log2<double> {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE void operator()(const Packet& x, Packet& log2_x_hi, Packet& log2_x_lo) {
// We use a transformation of variables:
// r = c * (x-1) / (x+1),
// such that
// log2(x) = log2((1 + r/c) / (1 - r/c)) = f(r).
// The function f(r) can be approximated well using an odd polynomial
// of the form
// P(r) = ((Q(r^2) * r^2 + C) * r^2 + 1) * r,
// For the implementation of log2<double> here, Q is of degree 6 with
// coefficient represented in working precision (double), while C is a
// constant represented in extra precision as a double word to achieve
// full accuracy.
//
// The polynomial coefficients were computed by the Sollya script:
//
// c = 2 / log(2);
// trans = c * (x-1)/(x+1);
// itrans = (1+x/c)/(1-x/c);
// interval=[trans(sqrt(0.5)); trans(sqrt(2))];
// print(interval);
// f = log2(itrans(x));
// p=fpminimax(f,[|1,3,5,7,9,11,13,15,17|],[|1,DD,double...|],interval,relative,floating);
const Packet q12 = pset1<Packet>(2.87074255468000586e-9);
const Packet q10 = pset1<Packet>(2.38957980901884082e-8);
const Packet q8 = pset1<Packet>(2.31032094540014656e-7);
const Packet q6 = pset1<Packet>(2.27279857398537278e-6);
const Packet q4 = pset1<Packet>(2.31271023278625638e-5);
const Packet q2 = pset1<Packet>(2.47556738444535513e-4);
const Packet q0 = pset1<Packet>(2.88543873228900172e-3);
const Packet C_hi = pset1<Packet>(0.0400377511598501157);
const Packet C_lo = pset1<Packet>(-4.77726582251425391e-19);
const Packet one = pset1<Packet>(1.0);
const Packet cst_2_log2e_hi = pset1<Packet>(2.88539008177792677);
const Packet cst_2_log2e_lo = pset1<Packet>(4.07660016854549667e-17);
// c * (x - 1)
Packet t_hi, t_lo;
// t = c * (x-1)
twoprod(cst_2_log2e_hi, cst_2_log2e_lo, psub(x, one), t_hi, t_lo);
// r = c * (x-1) / (x+1),
Packet r_hi, r_lo;
doubleword_div_fp(t_hi, t_lo, padd(x, one), r_hi, r_lo);
// r2 = r * r
Packet r2_hi, r2_lo;
twoprod(r_hi, r_lo, r_hi, r_lo, r2_hi, r2_lo);
// r4 = r2 * r2
Packet r4_hi, r4_lo;
twoprod(r2_hi, r2_lo, r2_hi, r2_lo, r4_hi, r4_lo);
// Evaluate Q(r^2) in working precision. We evaluate it in two parts
// (even and odd in r^2) to improve instruction level parallelism.
Packet q_even = pmadd(q12, r4_hi, q8);
Packet q_odd = pmadd(q10, r4_hi, q6);
q_even = pmadd(q_even, r4_hi, q4);
q_odd = pmadd(q_odd, r4_hi, q2);
q_even = pmadd(q_even, r4_hi, q0);
Packet q = pmadd(q_odd, r2_hi, q_even);
// Now evaluate the low order terms of P(x) in double word precision.
// In the following, due to the increasing magnitude of the coefficients
// and r being constrained to [-0.5, 0.5] we can use fast_twosum instead
// of the slower twosum.
// Q(r^2) * r^2
Packet p_hi, p_lo;
twoprod(r2_hi, r2_lo, q, p_hi, p_lo);
// Q(r^2) * r^2 + C
Packet p1_hi, p1_lo;
fast_twosum(C_hi, C_lo, p_hi, p_lo, p1_hi, p1_lo);
// (Q(r^2) * r^2 + C) * r^2
Packet p2_hi, p2_lo;
twoprod(r2_hi, r2_lo, p1_hi, p1_lo, p2_hi, p2_lo);
// ((Q(r^2) * r^2 + C) * r^2 + 1)
Packet p3_hi, p3_lo;
fast_twosum(one, p2_hi, p2_lo, p3_hi, p3_lo);
// log(z) ~= ((Q(r^2) * r^2 + C) * r^2 + 1) * r
twoprod(p3_hi, p3_lo, r_hi, r_lo, log2_x_hi, log2_x_lo);
}
};
// This function accurately computes exp2(x) for x in [-0.5:0.5], which is
// needed in pow(x,y).
template <typename Scalar>
struct fast_accurate_exp2 {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet operator()(const Packet& x) {
return generic_exp2(x);
}
};
// This specialization uses a faster algorithm to compute exp2(x) for floats
// in [-0.5;0.5] with a relative accuracy of 1 ulp.
// The minimax polynomial used was calculated using the Sollya tool.
// See sollya.org.
template <>
struct fast_accurate_exp2<float> {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet operator()(const Packet& x) {
// This function approximates exp2(x) by a degree 6 polynomial of the form
// Q(x) = 1 + x * (C + x * P(x)), where the degree 4 polynomial P(x) is evaluated in
// single precision, and the remaining steps are evaluated with extra precision using
// double word arithmetic. C is an extra precise constant stored as a double word.
//
// The polynomial coefficients were calculated using Sollya commands:
// > n = 6;
// > f = 2^x;
// > interval = [-0.5;0.5];
// > p = fpminimax(f,n,[|1,double,single...|],interval,relative,floating);
const Packet p4 = pset1<Packet>(1.539513905e-4f);
const Packet p3 = pset1<Packet>(1.340007293e-3f);
const Packet p2 = pset1<Packet>(9.618283249e-3f);
const Packet p1 = pset1<Packet>(5.550328270e-2f);
const Packet p0 = pset1<Packet>(0.2402264923f);
const Packet C_hi = pset1<Packet>(0.6931471825f);
const Packet C_lo = pset1<Packet>(2.36836577e-08f);
const Packet one = pset1<Packet>(1.0f);
// Evaluate P(x) in working precision.
// We evaluate even and odd parts of the polynomial separately
// to gain some instruction level parallelism.
Packet x2 = pmul(x, x);
Packet p_even = pmadd(p4, x2, p2);
Packet p_odd = pmadd(p3, x2, p1);
p_even = pmadd(p_even, x2, p0);
Packet p = pmadd(p_odd, x, p_even);
// Evaluate the remaining terms of Q(x) with extra precision using
// double word arithmetic.
Packet p_hi, p_lo;
// x * p(x)
twoprod(p, x, p_hi, p_lo);
// C + x * p(x)
Packet q1_hi, q1_lo;
twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo);
// x * (C + x * p(x))
Packet q2_hi, q2_lo;
twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo);
// 1 + x * (C + x * p(x))
Packet q3_hi, q3_lo;
// Since |q2_hi| <= sqrt(2)-1 < 1, we can use fast_twosum
// for adding it to unity here.
fast_twosum(one, q2_hi, q3_hi, q3_lo);
return padd(q3_hi, padd(q2_lo, q3_lo));
}
};
// in [-0.5;0.5] with a relative accuracy of 1 ulp.
// The minimax polynomial used was calculated using the Sollya tool.
// See sollya.org.
template <>
struct fast_accurate_exp2<double> {
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet operator()(const Packet& x) {
// This function approximates exp2(x) by a degree 10 polynomial of the form
// Q(x) = 1 + x * (C + x * P(x)), where the degree 8 polynomial P(x) is evaluated in
// single precision, and the remaining steps are evaluated with extra precision using
// double word arithmetic. C is an extra precise constant stored as a double word.
//
// The polynomial coefficients were calculated using Sollya commands:
// > n = 11;
// > f = 2^x;
// > interval = [-0.5;0.5];
// > p = fpminimax(f,n,[|1,DD,double...|],interval,relative,floating);
const Packet p9 = pset1<Packet>(4.431642109085495276e-10);
const Packet p8 = pset1<Packet>(7.073829923303358410e-9);
const Packet p7 = pset1<Packet>(1.017822306737031311e-7);
const Packet p6 = pset1<Packet>(1.321543498017646657e-6);
const Packet p5 = pset1<Packet>(1.525273342728892877e-5);
const Packet p4 = pset1<Packet>(1.540353045780084423e-4);
const Packet p3 = pset1<Packet>(1.333355814685869807e-3);
const Packet p2 = pset1<Packet>(9.618129107593478832e-3);
const Packet p1 = pset1<Packet>(5.550410866481961247e-2);
const Packet p0 = pset1<Packet>(0.240226506959101332);
const Packet C_hi = pset1<Packet>(0.693147180559945286);
const Packet C_lo = pset1<Packet>(4.81927865669806721e-17);
const Packet one = pset1<Packet>(1.0);
// Evaluate P(x) in working precision.
// We evaluate even and odd parts of the polynomial separately
// to gain some instruction level parallelism.
Packet x2 = pmul(x, x);
Packet p_even = pmadd(p8, x2, p6);
Packet p_odd = pmadd(p9, x2, p7);
p_even = pmadd(p_even, x2, p4);
p_odd = pmadd(p_odd, x2, p5);
p_even = pmadd(p_even, x2, p2);
p_odd = pmadd(p_odd, x2, p3);
p_even = pmadd(p_even, x2, p0);
p_odd = pmadd(p_odd, x2, p1);
Packet p = pmadd(p_odd, x, p_even);
// Evaluate the remaining terms of Q(x) with extra precision using
// double word arithmetic.
Packet p_hi, p_lo;
// x * p(x)
twoprod(p, x, p_hi, p_lo);
// C + x * p(x)
Packet q1_hi, q1_lo;
twosum(p_hi, p_lo, C_hi, C_lo, q1_hi, q1_lo);
// x * (C + x * p(x))
Packet q2_hi, q2_lo;
twoprod(q1_hi, q1_lo, x, q2_hi, q2_lo);
// 1 + x * (C + x * p(x))
Packet q3_hi, q3_lo;
// Since |q2_hi| <= sqrt(2)-1 < 1, we can use fast_twosum
// for adding it to unity here.
fast_twosum(one, q2_hi, q3_hi, q3_lo);
return padd(q3_hi, padd(q2_lo, q3_lo));
}
};
// This function implements the non-trivial case of pow(x,y) where x is
// positive and y is (possibly) non-integer.
// Formally, pow(x,y) = exp2(y * log2(x)), where exp2(x) is shorthand for 2^x.
// TODO(rmlarsen): We should probably add this as a packet up 'ppow', to make it
// easier to specialize or turn off for specific types and/or backends.x
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_pow_impl(const Packet& x, const Packet& y) {
typedef typename unpacket_traits<Packet>::type Scalar;
// Split x into exponent e_x and mantissa m_x.
Packet e_x;
Packet m_x = pfrexp(x, e_x);
// Adjust m_x to lie in [1/sqrt(2):sqrt(2)] to minimize absolute error in log2(m_x).
EIGEN_CONSTEXPR Scalar sqrt_half = Scalar(0.70710678118654752440);
const Packet m_x_scale_mask = pcmp_lt(m_x, pset1<Packet>(sqrt_half));
m_x = pselect(m_x_scale_mask, pmul(pset1<Packet>(Scalar(2)), m_x), m_x);
e_x = pselect(m_x_scale_mask, psub(e_x, pset1<Packet>(Scalar(1))), e_x);
// Compute log2(m_x) with 6 extra bits of accuracy.
Packet rx_hi, rx_lo;
accurate_log2<Scalar>()(m_x, rx_hi, rx_lo);
// Compute the two terms {y * e_x, y * r_x} in f = y * log2(x) with doubled
// precision using double word arithmetic.
Packet f1_hi, f1_lo, f2_hi, f2_lo;
twoprod(e_x, y, f1_hi, f1_lo);
twoprod(rx_hi, rx_lo, y, f2_hi, f2_lo);
// Sum the two terms in f using double word arithmetic. We know
// that |e_x| > |log2(m_x)|, except for the case where e_x==0.
// This means that we can use fast_twosum(f1,f2).
// In the case e_x == 0, e_x * y = f1 = 0, so we don't lose any
// accuracy by violating the assumption of fast_twosum, because
// it's a no-op.
Packet f_hi, f_lo;
fast_twosum(f1_hi, f1_lo, f2_hi, f2_lo, f_hi, f_lo);
// Split f into integer and fractional parts.
Packet n_z, r_z;
absolute_split(f_hi, n_z, r_z);
r_z = padd(r_z, f_lo);
Packet n_r;
absolute_split(r_z, n_r, r_z);
n_z = padd(n_z, n_r);
// We now have an accurate split of f = n_z + r_z and can compute
// x^y = 2**{n_z + r_z) = exp2(r_z) * 2**{n_z}.
// Since r_z is in [-0.5;0.5], we compute the first factor to high accuracy
// using a specialized algorithm. Multiplication by the second factor can
// be done exactly using pldexp(), since it is an integer power of 2.
const Packet e_r = fast_accurate_exp2<Scalar>()(r_z);
return pldexp(e_r, n_z);
}
// Generic implementation of pow(x,y).
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_pow(const Packet& x, const Packet& y) {
typedef typename unpacket_traits<Packet>::type Scalar;
const Packet cst_pos_inf = pset1<Packet>(NumTraits<Scalar>::infinity());
const Packet cst_neg_inf = pset1<Packet>(-NumTraits<Scalar>::infinity());
const Packet cst_zero = pset1<Packet>(Scalar(0));
const Packet cst_one = pset1<Packet>(Scalar(1));
const Packet cst_nan = pset1<Packet>(NumTraits<Scalar>::quiet_NaN());
const Packet abs_x = pabs(x);
// Predicates for sign and magnitude of x.
const Packet abs_x_is_zero = pcmp_eq(abs_x, cst_zero);
const Packet x_has_signbit = psignbit(x);
const Packet x_is_neg = pandnot(x_has_signbit, abs_x_is_zero);
const Packet x_is_neg_zero = pand(x_has_signbit, abs_x_is_zero);
const Packet abs_x_is_inf = pcmp_eq(abs_x, cst_pos_inf);
const Packet abs_x_is_one = pcmp_eq(abs_x, cst_one);
const Packet abs_x_is_gt_one = pcmp_lt(cst_one, abs_x);
const Packet abs_x_is_lt_one = pcmp_lt(abs_x, cst_one);
const Packet x_is_one = pandnot(abs_x_is_one, x_is_neg);
const Packet x_is_neg_one = pand(abs_x_is_one, x_is_neg);
const Packet x_is_nan = pisnan(x);
// Predicates for sign and magnitude of y.
const Packet abs_y = pabs(y);
const Packet y_is_one = pcmp_eq(y, cst_one);
const Packet abs_y_is_zero = pcmp_eq(abs_y, cst_zero);
const Packet y_is_neg = pcmp_lt(y, cst_zero);
const Packet y_is_pos = pandnot(ptrue(y), por(abs_y_is_zero, y_is_neg));
const Packet y_is_nan = pisnan(y);
const Packet abs_y_is_inf = pcmp_eq(abs_y, cst_pos_inf);
EIGEN_CONSTEXPR Scalar huge_exponent =
(NumTraits<Scalar>::max_exponent() * Scalar(EIGEN_LN2)) / NumTraits<Scalar>::epsilon();
const Packet abs_y_is_huge = pcmp_le(pset1<Packet>(huge_exponent), pabs(y));
// Predicates for whether y is integer and/or even.
const Packet y_is_int = pcmp_eq(pfloor(y), y);
const Packet y_div_2 = pmul(y, pset1<Packet>(Scalar(0.5)));
const Packet y_is_even = pcmp_eq(pround(y_div_2), y_div_2);
// Predicates encoding special cases for the value of pow(x,y)
const Packet invalid_negative_x = pandnot(pandnot(pandnot(x_is_neg, abs_x_is_inf), y_is_int), abs_y_is_inf);
const Packet pow_is_nan = por(invalid_negative_x, por(x_is_nan, y_is_nan));
const Packet pow_is_one =
por(por(x_is_one, abs_y_is_zero), pand(x_is_neg_one, por(abs_y_is_inf, pandnot(y_is_even, invalid_negative_x))));
const Packet pow_is_zero = por(por(por(pand(abs_x_is_zero, y_is_pos), pand(abs_x_is_inf, y_is_neg)),
pand(pand(abs_x_is_lt_one, abs_y_is_huge), y_is_pos)),
pand(pand(abs_x_is_gt_one, abs_y_is_huge), y_is_neg));
const Packet pow_is_inf = por(por(por(pand(abs_x_is_zero, y_is_neg), pand(abs_x_is_inf, y_is_pos)),
pand(pand(abs_x_is_lt_one, abs_y_is_huge), y_is_neg)),
pand(pand(abs_x_is_gt_one, abs_y_is_huge), y_is_pos));
const Packet pow_is_neg_zero = pand(pandnot(y_is_int, y_is_even),
por(pand(y_is_neg, pand(abs_x_is_inf, x_is_neg)), pand(y_is_pos, x_is_neg_zero)));
const Packet inf_val =
pselect(pandnot(pand(por(pand(abs_x_is_inf, x_is_neg), pand(x_is_neg_zero, y_is_neg)), y_is_int), y_is_even),
cst_neg_inf, cst_pos_inf);
// General computation of pow(x,y) for positive x or negative x and integer y.
const Packet negate_pow_abs = pandnot(x_is_neg, y_is_even);
const Packet pow_abs = generic_pow_impl(abs_x, y);
return pselect(y_is_one, x,
pselect(pow_is_one, cst_one,
pselect(pow_is_nan, cst_nan,
pselect(pow_is_inf, inf_val,
pselect(pow_is_neg_zero, pnegate(cst_zero),
pselect(pow_is_zero, cst_zero,
pselect(negate_pow_abs, pnegate(pow_abs), pow_abs)))))));
}
namespace unary_pow {
template <typename ScalarExponent, bool IsInteger = NumTraits<ScalarExponent>::IsInteger>
struct exponent_helper {
using safe_abs_type = ScalarExponent;
static constexpr ScalarExponent one_half = ScalarExponent(0.5);
// these routines assume that exp is an integer stored as a floating point type
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ScalarExponent safe_abs(const ScalarExponent& exp) {
return numext::abs(exp);
}
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE bool is_odd(const ScalarExponent& exp) {
eigen_assert(((numext::isfinite)(exp) && exp == numext::floor(exp)) && "exp must be an integer");
ScalarExponent exp_div_2 = exp * one_half;
ScalarExponent floor_exp_div_2 = numext::floor(exp_div_2);
return exp_div_2 != floor_exp_div_2;
}
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE ScalarExponent floor_div_two(const ScalarExponent& exp) {
ScalarExponent exp_div_2 = exp * one_half;
return numext::floor(exp_div_2);
}
};
template <typename ScalarExponent>
struct exponent_helper<ScalarExponent, true> {
// if `exp` is a signed integer type, cast it to its unsigned counterpart to safely store its absolute value
// consider the (rare) case where `exp` is an int32_t: abs(-2147483648) != 2147483648
using safe_abs_type = typename numext::get_integer_by_size<sizeof(ScalarExponent)>::unsigned_type;
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE safe_abs_type safe_abs(const ScalarExponent& exp) {
ScalarExponent mask = numext::signbit(exp);
safe_abs_type result = safe_abs_type(exp ^ mask);
return result + safe_abs_type(ScalarExponent(1) & mask);
}
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE bool is_odd(const safe_abs_type& exp) {
return exp % safe_abs_type(2) != safe_abs_type(0);
}
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE safe_abs_type floor_div_two(const safe_abs_type& exp) {
return exp >> safe_abs_type(1);
}
};
template <typename Packet, typename ScalarExponent,
bool ReciprocateIfExponentIsNegative =
!NumTraits<typename unpacket_traits<Packet>::type>::IsInteger && NumTraits<ScalarExponent>::IsSigned>
struct reciprocate {
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent& exponent) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_pos_one = pset1<Packet>(Scalar(1));
return exponent < 0 ? pdiv(cst_pos_one, x) : x;
}
};
template <typename Packet, typename ScalarExponent>
struct reciprocate<Packet, ScalarExponent, false> {
// pdiv not defined, nor necessary for integer base types
// if the exponent is unsigned, then the exponent cannot be negative
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent&) { return x; }
};
template <typename Packet, typename ScalarExponent>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet int_pow(const Packet& x, const ScalarExponent& exponent) {
using Scalar = typename unpacket_traits<Packet>::type;
using ExponentHelper = exponent_helper<ScalarExponent>;
using AbsExponentType = typename ExponentHelper::safe_abs_type;
const Packet cst_pos_one = pset1<Packet>(Scalar(1));
if (exponent == ScalarExponent(0)) return cst_pos_one;
Packet result = reciprocate<Packet, ScalarExponent>::run(x, exponent);
Packet y = cst_pos_one;
AbsExponentType m = ExponentHelper::safe_abs(exponent);
while (m > 1) {
bool odd = ExponentHelper::is_odd(m);
if (odd) y = pmul(y, result);
result = pmul(result, result);
m = ExponentHelper::floor_div_two(m);
}
return pmul(y, result);
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet gen_pow(const Packet& x,
const typename unpacket_traits<Packet>::type& exponent) {
const Packet exponent_packet = pset1<Packet>(exponent);
return generic_pow_impl(x, exponent_packet);
}
template <typename Packet, typename ScalarExponent>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet handle_nonint_nonint_errors(const Packet& x, const Packet& powx,
const ScalarExponent& exponent) {
using Scalar = typename unpacket_traits<Packet>::type;
// non-integer base and exponent case
const Scalar pos_zero = Scalar(0);
const Scalar all_ones = ptrue<Scalar>(Scalar());
const Scalar pos_one = Scalar(1);
const Scalar pos_inf = NumTraits<Scalar>::infinity();
const Packet cst_pos_zero = pzero(x);
const Packet cst_pos_one = pset1<Packet>(pos_one);
const Packet cst_pos_inf = pset1<Packet>(pos_inf);
const bool exponent_is_not_fin = !(numext::isfinite)(exponent);
const bool exponent_is_neg = exponent < ScalarExponent(0);
const bool exponent_is_pos = exponent > ScalarExponent(0);
const Packet exp_is_not_fin = pset1<Packet>(exponent_is_not_fin ? all_ones : pos_zero);
const Packet exp_is_neg = pset1<Packet>(exponent_is_neg ? all_ones : pos_zero);
const Packet exp_is_pos = pset1<Packet>(exponent_is_pos ? all_ones : pos_zero);
const Packet exp_is_inf = pand(exp_is_not_fin, por(exp_is_neg, exp_is_pos));
const Packet exp_is_nan = pandnot(exp_is_not_fin, por(exp_is_neg, exp_is_pos));
const Packet x_is_le_zero = pcmp_le(x, cst_pos_zero);
const Packet x_is_ge_zero = pcmp_le(cst_pos_zero, x);
const Packet x_is_zero = pand(x_is_le_zero, x_is_ge_zero);
const Packet abs_x = pabs(x);
const Packet abs_x_is_le_one = pcmp_le(abs_x, cst_pos_one);
const Packet abs_x_is_ge_one = pcmp_le(cst_pos_one, abs_x);
const Packet abs_x_is_inf = pcmp_eq(abs_x, cst_pos_inf);
const Packet abs_x_is_one = pand(abs_x_is_le_one, abs_x_is_ge_one);
Packet pow_is_inf_if_exp_is_neg = por(x_is_zero, pand(abs_x_is_le_one, exp_is_inf));
Packet pow_is_inf_if_exp_is_pos = por(abs_x_is_inf, pand(abs_x_is_ge_one, exp_is_inf));
Packet pow_is_one = pand(abs_x_is_one, por(exp_is_inf, x_is_ge_zero));
Packet result = powx;
result = por(x_is_le_zero, result);
result = pselect(pow_is_inf_if_exp_is_neg, pand(cst_pos_inf, exp_is_neg), result);
result = pselect(pow_is_inf_if_exp_is_pos, pand(cst_pos_inf, exp_is_pos), result);
result = por(exp_is_nan, result);
result = pselect(pow_is_one, cst_pos_one, result);
return result;
}
template <typename Packet, typename ScalarExponent,
std::enable_if_t<NumTraits<typename unpacket_traits<Packet>::type>::IsSigned, bool> = true>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet handle_negative_exponent(const Packet& x, const ScalarExponent& exponent) {
using Scalar = typename unpacket_traits<Packet>::type;
// signed integer base, signed integer exponent case
// This routine handles negative exponents.
// The return value is either 0, 1, or -1.
const Scalar pos_zero = Scalar(0);
const Scalar all_ones = ptrue<Scalar>(Scalar());
const Scalar pos_one = Scalar(1);
const Packet cst_pos_one = pset1<Packet>(pos_one);
const bool exponent_is_odd = exponent % ScalarExponent(2) != ScalarExponent(0);
const Packet exp_is_odd = pset1<Packet>(exponent_is_odd ? all_ones : pos_zero);
const Packet abs_x = pabs(x);
const Packet abs_x_is_one = pcmp_eq(abs_x, cst_pos_one);
Packet result = pselect(exp_is_odd, x, abs_x);
result = pand(abs_x_is_one, result);
return result;
}
template <typename Packet, typename ScalarExponent,
std::enable_if_t<!NumTraits<typename unpacket_traits<Packet>::type>::IsSigned, bool> = true>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet handle_negative_exponent(const Packet& x, const ScalarExponent&) {
using Scalar = typename unpacket_traits<Packet>::type;
// unsigned integer base, signed integer exponent case
// This routine handles negative exponents.
// The return value is either 0 or 1
const Scalar pos_one = Scalar(1);
const Packet cst_pos_one = pset1<Packet>(pos_one);
const Packet x_is_one = pcmp_eq(x, cst_pos_one);
return pand(x_is_one, x);
}
} // end namespace unary_pow
template <typename Packet, typename ScalarExponent,
bool BaseIsIntegerType = NumTraits<typename unpacket_traits<Packet>::type>::IsInteger,
bool ExponentIsIntegerType = NumTraits<ScalarExponent>::IsInteger,
bool ExponentIsSigned = NumTraits<ScalarExponent>::IsSigned>
struct unary_pow_impl;
template <typename Packet, typename ScalarExponent, bool ExponentIsSigned>
struct unary_pow_impl<Packet, ScalarExponent, false, false, ExponentIsSigned> {
typedef typename unpacket_traits<Packet>::type Scalar;
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent& exponent) {
const bool exponent_is_integer = (numext::isfinite)(exponent) && numext::round(exponent) == exponent;
if (exponent_is_integer) {
return unary_pow::int_pow(x, exponent);
} else {
Packet result = unary_pow::gen_pow(x, exponent);
result = unary_pow::handle_nonint_nonint_errors(x, result, exponent);
return result;
}
}
};
template <typename Packet, typename ScalarExponent, bool ExponentIsSigned>
struct unary_pow_impl<Packet, ScalarExponent, false, true, ExponentIsSigned> {
typedef typename unpacket_traits<Packet>::type Scalar;
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent& exponent) {
return unary_pow::int_pow(x, exponent);
}
};
template <typename Packet, typename ScalarExponent>
struct unary_pow_impl<Packet, ScalarExponent, true, true, true> {
typedef typename unpacket_traits<Packet>::type Scalar;
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent& exponent) {
if (exponent < ScalarExponent(0)) {
return unary_pow::handle_negative_exponent(x, exponent);
} else {
return unary_pow::int_pow(x, exponent);
}
}
};
template <typename Packet, typename ScalarExponent>
struct unary_pow_impl<Packet, ScalarExponent, true, true, false> {
typedef typename unpacket_traits<Packet>::type Scalar;
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run(const Packet& x, const ScalarExponent& exponent) {
return unary_pow::int_pow(x, exponent);
}
};
// This function computes exp2(x) = exp(ln(2) * x).
// To improve accuracy, the product ln(2)*x is computed using the twoprod
// algorithm, such that ln(2) * x = p_hi + p_lo holds exactly. Then exp2(x) is
// computed as exp2(x) = exp(p_hi) * exp(p_lo) ~= exp(p_hi) * (1 + p_lo). This
// correction step this reduces the maximum absolute error as follows:
//
// type | max error (simple product) | max error (twoprod) |
// -----------------------------------------------------------
// float | 35 ulps | 4 ulps |
// double | 363 ulps | 110 ulps |
//
template <typename Packet>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS Packet generic_exp2(const Packet& _x) {
typedef typename unpacket_traits<Packet>::type Scalar;
constexpr int max_exponent = std::numeric_limits<Scalar>::max_exponent;
constexpr int digits = std::numeric_limits<Scalar>::digits;
constexpr Scalar max_cap = Scalar(max_exponent + 1);
constexpr Scalar min_cap = -Scalar(max_exponent + digits - 1);
Packet x = pmax(pmin(_x, pset1<Packet>(max_cap)), pset1<Packet>(min_cap));
Packet p_hi, p_lo;
twoprod(pset1<Packet>(Scalar(EIGEN_LN2)), x, p_hi, p_lo);
Packet exp2_hi = pexp(p_hi);
Packet exp2_lo = padd(pset1<Packet>(Scalar(1)), p_lo);
return pmul(exp2_hi, exp2_lo);
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_rint(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
using IntType = typename numext::get_integer_by_size<sizeof(Scalar)>::signed_type;
// Adds and subtracts signum(a) * 2^kMantissaBits to force rounding.
const IntType kLimit = IntType(1) << (NumTraits<Scalar>::digits() - 1);
const Packet cst_limit = pset1<Packet>(static_cast<Scalar>(kLimit));
Packet abs_a = pabs(a);
Packet sign_a = pandnot(a, abs_a);
Packet rint_a = padd(abs_a, cst_limit);
// Don't compile-away addition and subtraction.
EIGEN_OPTIMIZATION_BARRIER(rint_a);
rint_a = psub(rint_a, cst_limit);
rint_a = por(rint_a, sign_a);
// If greater than limit (or NaN), simply return a.
Packet mask = pcmp_lt(abs_a, cst_limit);
Packet result = pselect(mask, rint_a, a);
return result;
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_floor(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_1 = pset1<Packet>(Scalar(1));
Packet rint_a = generic_rint(a);
// if a < rint(a), then rint(a) == ceil(a)
Packet mask = pcmp_lt(a, rint_a);
Packet offset = pand(cst_1, mask);
Packet result = psub(rint_a, offset);
return result;
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_ceil(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_1 = pset1<Packet>(Scalar(1));
const Packet sign_mask = pset1<Packet>(static_cast<Scalar>(-0.0));
Packet rint_a = generic_rint(a);
// if rint(a) < a, then rint(a) == floor(a)
Packet mask = pcmp_lt(rint_a, a);
Packet offset = pand(cst_1, mask);
Packet result = padd(rint_a, offset);
// Signed zero must remain signed (e.g. ceil(-0.02) == -0).
result = por(result, pand(sign_mask, a));
return result;
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_trunc(const Packet& a) {
Packet abs_a = pabs(a);
Packet sign_a = pandnot(a, abs_a);
Packet floor_abs_a = generic_floor(abs_a);
Packet result = por(floor_abs_a, sign_a);
return result;
}
template <typename Packet>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet generic_round(const Packet& a) {
using Scalar = typename unpacket_traits<Packet>::type;
const Packet cst_half = pset1<Packet>(Scalar(0.5));
const Packet cst_1 = pset1<Packet>(Scalar(1));
Packet abs_a = pabs(a);
Packet sign_a = pandnot(a, abs_a);
Packet floor_abs_a = generic_floor(abs_a);
Packet diff = psub(abs_a, floor_abs_a);
Packet mask = pcmp_le(cst_half, diff);
Packet offset = pand(cst_1, mask);
Packet result = padd(floor_abs_a, offset);
result = por(result, sign_a);
return result;
}
template <typename Packet>
struct nearest_integer_packetop_impl<Packet, /*IsScalar*/ false, /*IsInteger*/ false> {
using Scalar = typename unpacket_traits<Packet>::type;
static_assert(packet_traits<Scalar>::HasRound, "Generic nearest integer functions are disabled for this type.");
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_floor(const Packet& x) { return generic_floor(x); }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_ceil(const Packet& x) { return generic_ceil(x); }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_rint(const Packet& x) { return generic_rint(x); }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_round(const Packet& x) { return generic_round(x); }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_trunc(const Packet& x) { return generic_trunc(x); }
};
template <typename Packet>
struct nearest_integer_packetop_impl<Packet, /*IsScalar*/ false, /*IsInteger*/ true> {
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_floor(const Packet& x) { return x; }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_ceil(const Packet& x) { return x; }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_rint(const Packet& x) { return x; }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_round(const Packet& x) { return x; }
static EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE Packet run_trunc(const Packet& x) { return x; }
};
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_ARCH_GENERIC_PACKET_MATH_FUNCTIONS_H