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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
//
// 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/.
#ifndef EIGEN_MATH_FUNCTIONS_AVX_H
#define EIGEN_MATH_FUNCTIONS_AVX_H
/* The sin, cos, exp, and log functions of this file are loosely derived from
* Julien Pommier's sse math library: http://gruntthepeon.free.fr/ssemath/
*/
namespace Eigen {
namespace internal {
inline Packet8i pshiftleft(Packet8i v, int n)
{
#ifdef EIGEN_VECTORIZE_AVX2
return _mm256_slli_epi32(v, n);
#else
__m128i lo = _mm_slli_epi32(_mm256_extractf128_si256(v, 0), n);
__m128i hi = _mm_slli_epi32(_mm256_extractf128_si256(v, 1), n);
return _mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1);
#endif
}
inline Packet8f pshiftright(Packet8f v, int n)
{
#ifdef EIGEN_VECTORIZE_AVX2
return _mm256_cvtepi32_ps(_mm256_srli_epi32(_mm256_castps_si256(v), n));
#else
__m128i lo = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(v), 0), n);
__m128i hi = _mm_srli_epi32(_mm256_extractf128_si256(_mm256_castps_si256(v), 1), n);
return _mm256_cvtepi32_ps(_mm256_insertf128_si256(_mm256_castsi128_si256(lo), (hi), 1));
#endif
}
// Sine function
// Computes sin(x) by wrapping x to the interval [-Pi/4,3*Pi/4] and
// evaluating interpolants in [-Pi/4,Pi/4] or [Pi/4,3*Pi/4]. The interpolants
// are (anti-)symmetric and thus have only odd/even coefficients
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
psin<Packet8f>(const Packet8f& _x) {
Packet8f x = _x;
// Some useful values.
_EIGEN_DECLARE_CONST_Packet8i(one, 1);
_EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
_EIGEN_DECLARE_CONST_Packet8f(two, 2.0f);
_EIGEN_DECLARE_CONST_Packet8f(one_over_four, 0.25f);
_EIGEN_DECLARE_CONST_Packet8f(one_over_pi, 3.183098861837907e-01f);
_EIGEN_DECLARE_CONST_Packet8f(neg_pi_first, -3.140625000000000e+00f);
_EIGEN_DECLARE_CONST_Packet8f(neg_pi_second, -9.670257568359375e-04f);
_EIGEN_DECLARE_CONST_Packet8f(neg_pi_third, -6.278329571784980e-07f);
_EIGEN_DECLARE_CONST_Packet8f(four_over_pi, 1.273239544735163e+00f);
// Map x from [-Pi/4,3*Pi/4] to z in [-1,3] and subtract the shifted period.
Packet8f z = pmul(x, p8f_one_over_pi);
Packet8f shift = _mm256_floor_ps(padd(z, p8f_one_over_four));
x = pmadd(shift, p8f_neg_pi_first, x);
x = pmadd(shift, p8f_neg_pi_second, x);
x = pmadd(shift, p8f_neg_pi_third, x);
z = pmul(x, p8f_four_over_pi);
// Make a mask for the entries that need flipping, i.e. wherever the shift
// is odd.
Packet8i shift_ints = _mm256_cvtps_epi32(shift);
Packet8i shift_isodd = _mm256_castps_si256(_mm256_and_ps(_mm256_castsi256_ps(shift_ints), _mm256_castsi256_ps(p8i_one)));
Packet8i sign_flip_mask = pshiftleft(shift_isodd, 31);
// Create a mask for which interpolant to use, i.e. if z > 1, then the mask
// is set to ones for that entry.
Packet8f ival_mask = _mm256_cmp_ps(z, p8f_one, _CMP_GT_OQ);
// Evaluate the polynomial for the interval [1,3] in z.
_EIGEN_DECLARE_CONST_Packet8f(coeff_right_0, 9.999999724233232e-01f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_right_2, -3.084242535619928e-01f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_right_4, 1.584991525700324e-02f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_right_6, -3.188805084631342e-04f);
Packet8f z_minus_two = psub(z, p8f_two);
Packet8f z_minus_two2 = pmul(z_minus_two, z_minus_two);
Packet8f right = pmadd(p8f_coeff_right_6, z_minus_two2, p8f_coeff_right_4);
right = pmadd(right, z_minus_two2, p8f_coeff_right_2);
right = pmadd(right, z_minus_two2, p8f_coeff_right_0);
// Evaluate the polynomial for the interval [-1,1] in z.
_EIGEN_DECLARE_CONST_Packet8f(coeff_left_1, 7.853981525427295e-01f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_left_3, -8.074536727092352e-02f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_left_5, 2.489871967827018e-03f);
_EIGEN_DECLARE_CONST_Packet8f(coeff_left_7, -3.587725841214251e-05f);
Packet8f z2 = pmul(z, z);
Packet8f left = pmadd(p8f_coeff_left_7, z2, p8f_coeff_left_5);
left = pmadd(left, z2, p8f_coeff_left_3);
left = pmadd(left, z2, p8f_coeff_left_1);
left = pmul(left, z);
// Assemble the results, i.e. select the left and right polynomials.
left = _mm256_andnot_ps(ival_mask, left);
right = _mm256_and_ps(ival_mask, right);
Packet8f res = _mm256_or_ps(left, right);
// Flip the sign on the odd intervals and return the result.
res = _mm256_xor_ps(res, _mm256_castsi256_ps(sign_flip_mask));
return res;
}
// Natural 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 <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
plog<Packet8f>(const Packet8f& _x) {
Packet8f x = _x;
_EIGEN_DECLARE_CONST_Packet8f(1, 1.0f);
_EIGEN_DECLARE_CONST_Packet8f(half, 0.5f);
_EIGEN_DECLARE_CONST_Packet8f(126f, 126.0f);
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inv_mant_mask, ~0x7f800000);
// The smallest non denormalized float number.
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(min_norm_pos, 0x00800000);
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(minus_inf, 0xff800000);
// Polynomial coefficients.
_EIGEN_DECLARE_CONST_Packet8f(cephes_SQRTHF, 0.707106781186547524f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p0, 7.0376836292E-2f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p1, -1.1514610310E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p2, 1.1676998740E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p3, -1.2420140846E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p4, +1.4249322787E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p5, -1.6668057665E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p6, +2.0000714765E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p7, -2.4999993993E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_p8, +3.3333331174E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_q1, -2.12194440e-4f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_log_q2, 0.693359375f);
Packet8f invalid_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_NGE_UQ); // not greater equal is true if x is NaN
Packet8f iszero_mask = _mm256_cmp_ps(x, _mm256_setzero_ps(), _CMP_EQ_OQ);
// Truncate input values to the minimum positive normal.
x = pmax(x, p8f_min_norm_pos);
Packet8f emm0 = pshiftright(x,23);
Packet8f e = _mm256_sub_ps(emm0, p8f_126f);
// Set the exponents to -1, i.e. x are in the range [0.5,1).
x = _mm256_and_ps(x, p8f_inv_mant_mask);
x = _mm256_or_ps(x, p8f_half);
// 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; }
Packet8f mask = _mm256_cmp_ps(x, p8f_cephes_SQRTHF, _CMP_LT_OQ);
Packet8f tmp = _mm256_and_ps(x, mask);
x = psub(x, p8f_1);
e = psub(e, _mm256_and_ps(p8f_1, mask));
x = padd(x, tmp);
Packet8f x2 = pmul(x, x);
Packet8f x3 = pmul(x2, x);
// Evaluate the polynomial approximant of degree 8 in three parts, probably
// to improve instruction-level parallelism.
Packet8f y, y1, y2;
y = pmadd(p8f_cephes_log_p0, x, p8f_cephes_log_p1);
y1 = pmadd(p8f_cephes_log_p3, x, p8f_cephes_log_p4);
y2 = pmadd(p8f_cephes_log_p6, x, p8f_cephes_log_p7);
y = pmadd(y, x, p8f_cephes_log_p2);
y1 = pmadd(y1, x, p8f_cephes_log_p5);
y2 = pmadd(y2, x, p8f_cephes_log_p8);
y = pmadd(y, x3, y1);
y = pmadd(y, x3, y2);
y = pmul(y, x3);
// Add the logarithm of the exponent back to the result of the interpolation.
y1 = pmul(e, p8f_cephes_log_q1);
tmp = pmul(x2, p8f_half);
y = padd(y, y1);
x = psub(x, tmp);
y2 = pmul(e, p8f_cephes_log_q2);
x = padd(x, y);
x = padd(x, y2);
// Filter out invalid inputs, i.e. negative arg will be NAN, 0 will be -INF.
return _mm256_or_ps(
_mm256_andnot_ps(iszero_mask, _mm256_or_ps(x, invalid_mask)),
_mm256_and_ps(iszero_mask, p8f_minus_inf));
}
// 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).
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
pexp<Packet8f>(const Packet8f& _x) {
_EIGEN_DECLARE_CONST_Packet8f(1, 1.0f);
_EIGEN_DECLARE_CONST_Packet8f(half, 0.5f);
_EIGEN_DECLARE_CONST_Packet8f(127, 127.0f);
_EIGEN_DECLARE_CONST_Packet8f(exp_hi, 88.3762626647950f);
_EIGEN_DECLARE_CONST_Packet8f(exp_lo, -88.3762626647949f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_LOG2EF, 1.44269504088896341f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p0, 1.9875691500E-4f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p1, 1.3981999507E-3f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p2, 8.3334519073E-3f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p3, 4.1665795894E-2f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p4, 1.6666665459E-1f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_p5, 5.0000001201E-1f);
// Clamp x.
Packet8f x = pmax(pmin(_x, p8f_exp_hi), p8f_exp_lo);
// Express exp(x) as exp(m*ln(2) + r), start by extracting
// m = floor(x/ln(2) + 0.5).
Packet8f m = _mm256_floor_ps(pmadd(x, p8f_cephes_LOG2EF, p8f_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. Note that we don't use the "pmadd" function here to
// ensure that a precision-preserving FMA instruction is used.
#ifdef EIGEN_VECTORIZE_FMA
_EIGEN_DECLARE_CONST_Packet8f(nln2, -0.6931471805599453f);
Packet8f r = _mm256_fmadd_ps(m, p8f_nln2, x);
#else
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_C1, 0.693359375f);
_EIGEN_DECLARE_CONST_Packet8f(cephes_exp_C2, -2.12194440e-4f);
Packet8f r = psub(x, pmul(m, p8f_cephes_exp_C1));
r = psub(r, pmul(m, p8f_cephes_exp_C2));
#endif
Packet8f r2 = pmul(r, r);
// TODO(gonnet): Split into odd/even polynomials and try to exploit
// instruction-level parallelism.
Packet8f y = p8f_cephes_exp_p0;
y = pmadd(y, r, p8f_cephes_exp_p1);
y = pmadd(y, r, p8f_cephes_exp_p2);
y = pmadd(y, r, p8f_cephes_exp_p3);
y = pmadd(y, r, p8f_cephes_exp_p4);
y = pmadd(y, r, p8f_cephes_exp_p5);
y = pmadd(y, r2, r);
y = padd(y, p8f_1);
// Build emm0 = 2^m.
Packet8i emm0 = _mm256_cvttps_epi32(padd(m, p8f_127));
emm0 = pshiftleft(emm0, 23);
// Return 2^m * exp(r).
return pmax(pmul(y, _mm256_castsi256_ps(emm0)), _x);
}
// Hyperbolic Tangent function.
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
ptanh<Packet8f>(const Packet8f& x) {
return internal::generic_fast_tanh_float(x);
}
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet4d
pexp<Packet4d>(const Packet4d& _x) {
Packet4d x = _x;
_EIGEN_DECLARE_CONST_Packet4d(1, 1.0);
_EIGEN_DECLARE_CONST_Packet4d(2, 2.0);
_EIGEN_DECLARE_CONST_Packet4d(half, 0.5);
_EIGEN_DECLARE_CONST_Packet4d(exp_hi, 709.437);
_EIGEN_DECLARE_CONST_Packet4d(exp_lo, -709.436139303);
_EIGEN_DECLARE_CONST_Packet4d(cephes_LOG2EF, 1.4426950408889634073599);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p0, 1.26177193074810590878e-4);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p1, 3.02994407707441961300e-2);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_p2, 9.99999999999999999910e-1);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q0, 3.00198505138664455042e-6);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q1, 2.52448340349684104192e-3);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q2, 2.27265548208155028766e-1);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_q3, 2.00000000000000000009e0);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C1, 0.693145751953125);
_EIGEN_DECLARE_CONST_Packet4d(cephes_exp_C2, 1.42860682030941723212e-6);
_EIGEN_DECLARE_CONST_Packet4i(1023, 1023);
Packet4d tmp, fx;
// clamp x
x = pmax(pmin(x, p4d_exp_hi), p4d_exp_lo);
// Express exp(x) as exp(g + n*log(2)).
fx = pmadd(p4d_cephes_LOG2EF, x, p4d_half);
// Get the integer modulus of log(2), i.e. the "n" described above.
fx = _mm256_floor_pd(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, p4d_cephes_exp_C1);
Packet4d z = pmul(fx, p4d_cephes_exp_C2);
x = psub(x, tmp);
x = psub(x, z);
Packet4d x2 = pmul(x, x);
// Evaluate the numerator polynomial of the rational interpolant.
Packet4d px = p4d_cephes_exp_p0;
px = pmadd(px, x2, p4d_cephes_exp_p1);
px = pmadd(px, x2, p4d_cephes_exp_p2);
px = pmul(px, x);
// Evaluate the denominator polynomial of the rational interpolant.
Packet4d qx = p4d_cephes_exp_q0;
qx = pmadd(qx, x2, p4d_cephes_exp_q1);
qx = pmadd(qx, x2, p4d_cephes_exp_q2);
qx = pmadd(qx, x2, p4d_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 = _mm256_div_pd(px, psub(qx, px));
x = pmadd(p4d_2, x, p4d_1);
// Build e=2^n by constructing the exponents in a 128-bit vector and
// shifting them to where they belong in double-precision values.
__m128i emm0 = _mm256_cvtpd_epi32(fx);
emm0 = _mm_add_epi32(emm0, p4i_1023);
emm0 = _mm_shuffle_epi32(emm0, _MM_SHUFFLE(3, 1, 2, 0));
__m128i lo = _mm_slli_epi64(emm0, 52);
__m128i hi = _mm_slli_epi64(_mm_srli_epi64(emm0, 32), 52);
__m256i e = _mm256_insertf128_si256(_mm256_setzero_si256(), lo, 0);
e = _mm256_insertf128_si256(e, hi, 1);
// Construct the result 2^n * exp(g) = e * x. The max is used to catch
// non-finite values in the input.
return pmax(pmul(x, _mm256_castsi256_pd(e)), _x);
}
// Functions for sqrt.
// The EIGEN_FAST_MATH version uses the _mm_rsqrt_ps approximation and one step
// of Newton's method, at a cost of 1-2 bits of precision as opposed to the
// exact solution. It does not handle +inf, or denormalized numbers correctly.
// The main advantage of this approach is not just speed, but also the fact that
// it can be inlined and pipelined with other computations, further reducing its
// effective latency. This is similar to Quake3's fast inverse square root.
// For detail see here: http://www.beyond3d.com/content/articles/8/
#if EIGEN_FAST_MATH
template <>
EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED Packet8f
psqrt<Packet8f>(const Packet8f& _x) {
Packet8f half = pmul(_x, pset1<Packet8f>(.5f));
Packet8f denormal_mask = _mm256_and_ps(
_mm256_cmp_ps(_x, pset1<Packet8f>((std::numeric_limits<float>::min)()),
_CMP_LT_OQ),
_mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_GE_OQ));
// Compute approximate reciprocal sqrt.
Packet8f x = _mm256_rsqrt_ps(_x);
// Do a single step of Newton's iteration.
x = pmul(x, psub(pset1<Packet8f>(1.5f), pmul(half, pmul(x,x))));
// Flush results for denormals to zero.
return _mm256_andnot_ps(denormal_mask, pmul(_x,x));
}
#else
template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet8f psqrt<Packet8f>(const Packet8f& x) {
return _mm256_sqrt_ps(x);
}
#endif
template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4d psqrt<Packet4d>(const Packet4d& x) {
return _mm256_sqrt_pd(x);
}
#if EIGEN_FAST_MATH
template<> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet8f prsqrt<Packet8f>(const Packet8f& _x) {
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(inf, 0x7f800000);
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(nan, 0x7fc00000);
_EIGEN_DECLARE_CONST_Packet8f(one_point_five, 1.5f);
_EIGEN_DECLARE_CONST_Packet8f(minus_half, -0.5f);
_EIGEN_DECLARE_CONST_Packet8f_FROM_INT(flt_min, 0x00800000);
Packet8f neg_half = pmul(_x, p8f_minus_half);
// select only the inverse sqrt of positive normal inputs (denormals are
// flushed to zero and cause infs as well).
Packet8f le_zero_mask = _mm256_cmp_ps(_x, p8f_flt_min, _CMP_LT_OQ);
Packet8f x = _mm256_andnot_ps(le_zero_mask, _mm256_rsqrt_ps(_x));
// Fill in NaNs and Infs for the negative/zero entries.
Packet8f neg_mask = _mm256_cmp_ps(_x, _mm256_setzero_ps(), _CMP_LT_OQ);
Packet8f zero_mask = _mm256_andnot_ps(neg_mask, le_zero_mask);
Packet8f infs_and_nans = _mm256_or_ps(_mm256_and_ps(neg_mask, p8f_nan),
_mm256_and_ps(zero_mask, p8f_inf));
// Do a single step of Newton's iteration.
x = pmul(x, pmadd(neg_half, pmul(x, x), p8f_one_point_five));
// Insert NaNs and Infs in all the right places.
return _mm256_or_ps(x, infs_and_nans);
}
#else
template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet8f prsqrt<Packet8f>(const Packet8f& x) {
_EIGEN_DECLARE_CONST_Packet8f(one, 1.0f);
return _mm256_div_ps(p8f_one, _mm256_sqrt_ps(x));
}
#endif
template <> EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS EIGEN_UNUSED
Packet4d prsqrt<Packet4d>(const Packet4d& x) {
_EIGEN_DECLARE_CONST_Packet4d(one, 1.0);
return _mm256_div_pd(p4d_one, _mm256_sqrt_pd(x));
}
} // end namespace internal
} // end namespace Eigen
#endif // EIGEN_MATH_FUNCTIONS_AVX_H