From cdd8fdc32e730d5a65796a791ff13a92815c59b9 Mon Sep 17 00:00:00 2001 From: Rasmus Munk Larsen Date: Mon, 18 Jan 2021 13:25:16 +0000 Subject: Vectorize `pow(x, y)`. This closes https://gitlab.com/libeigen/eigen/-/issues/2085, which also contains a description of the algorithm. I ran some testing (comparing to `std::pow(double(x), double(y)))` for `x` in the set of all (positive) floats in the interval `[std::sqrt(std::numeric_limits::min()), std::sqrt(std::numeric_limits::max())]`, and `y` in `{2, sqrt(2), -sqrt(2)}` I get the following error statistics: ``` max_rel_error = 8.34405e-07 rms_rel_error = 2.76654e-07 ``` If I widen the range to all normal float I see lower accuracy for arguments where the result is subnormal, e.g. for `y = sqrt(2)`: ``` max_rel_error = 0.666667 rms = 6.8727e-05 count = 1335165689 argmax = 2.56049e-32, 2.10195e-45 != 1.4013e-45 ``` which seems reasonable, since these results are subnormals with only couple of significant bits left. --- .../Core/arch/Default/GenericPacketMathFunctions.h | 165 +++++++++++++++++++++ 1 file changed, 165 insertions(+) (limited to 'Eigen/src/Core/arch/Default/GenericPacketMathFunctions.h') diff --git a/Eigen/src/Core/arch/Default/GenericPacketMathFunctions.h b/Eigen/src/Core/arch/Default/GenericPacketMathFunctions.h index f40093455..9a1feb0d9 100644 --- a/Eigen/src/Core/arch/Default/GenericPacketMathFunctions.h +++ b/Eigen/src/Core/arch/Default/GenericPacketMathFunctions.h @@ -793,6 +793,171 @@ Packet psqrt_complex(const Packet& a) { pselect(is_real_inf, real_inf_result,result)); } + +// 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 code corresponds to Algorithms 3 and 4 in +// https://hal.inria.fr/hal-01774587v2/document +template +EIGEN_STRONG_INLINE +void veltkamp_splitting(const Packet& x, Packet& x_hi, Packet& x_lo) { + typedef typename unpacket_traits::type Scalar; + EIGEN_CONSTEXPR int shift = (NumTraits::digits() + 1) / 2; + EIGEN_CONSTEXPR Scalar shift_scale = Scalar(uint64_t(1) << shift); + Packet gamma = pmul(pset1(shift_scale + 1), x); +#ifdef EIGEN_HAS_SINGLE_INSTRUCTION_MADD + x_hi = pmadd(pset1(-shift_scale), x, gamma); +#else + Packet rho = psub(x, gamma); + x_hi = padd(rho, gamma); +#endif + x_lo = psub(x, x_hi); +} + +// This function splits x into the nearest integer n and fractional part r, +// such that x = n + r holds exactly. +template +EIGEN_STRONG_INLINE +void integer_split(const Packet& x, Packet& n, Packet& r) { + n = pround(x); + r = psub(x, n); +} + +// This function implements Dekker's algorithm for two products {x * y1, x * y2} with +// a shared factor. Given floating point numbers {x, y1, y2} computes the pairs +// {p1, r1} and {p2, r2} such that x * y1 = p1 + r1 holds exactly and +// p1 = fl(x * y1), and x * y2 = p2 + r2 holds exactly and p2 = fl(x * y2). +template +EIGEN_STRONG_INLINE +void double_dekker(const Packet& x, const Packet& y1, const Packet& y2, + Packet& p1, Packet& r1, Packet& p2, Packet& r2) { + Packet x_hi, x_lo, y1_hi, y1_lo, y2_hi, y2_lo; + veltkamp_splitting(x, x_hi, x_lo); + veltkamp_splitting(y1, y1_hi, y1_lo); + veltkamp_splitting(y2, y2_hi, y2_lo); + + p1 = pmul(x, y1); + r1 = pmadd(x_hi, y1_hi, pnegate(p1)); + r1 = pmadd(x_hi, y1_lo, r1); + r1 = pmadd(x_lo, y1_hi, r1); + r1 = pmadd(x_lo, y1_lo, r1); + + p2 = pmul(x, y2); + r2 = pmadd(x_hi, y2_hi, pnegate(p2)); + r2 = pmadd(x_hi, y2_lo, r2); + r2 = pmadd(x_lo, y2_hi, r2); + r2 = pmadd(x_lo, y2_lo, r2); +} + +// 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) = 2**(y * log2(x)) +template +EIGEN_STRONG_INLINE +Packet generic_pow_impl(const Packet& x, const Packet& y) { + typedef typename unpacket_traits::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 [0.75:1.5) to minimize absolute error in log2(m_x). + Packet m_x_scale_mask = pcmp_lt(m_x, pset1(Scalar(0.75))); + m_x = pselect(m_x_scale_mask, pmul(pset1(Scalar(2)), m_x), m_x); + e_x = pselect(m_x_scale_mask, psub(e_x, pset1(Scalar(1))), e_x); + + Packet r_x = plog2(m_x); + + // Compute the two terms {y * e_x, y * r_x} in f = y * log2(x) with doubled + // precision using Dekker's algorithm. + Packet f1_hi, f1_lo, f2_hi, f2_lo; + double_dekker(y, e_x, r_x, f1_hi, f1_lo, f2_hi, f2_lo); + + // Separate f into integer and fractional parts, keeping f1_hi, and f2_hi + // separate to avoid cancellation. + Packet n1, r1, n2, r2; + integer_split(f1_hi, n1, r1); + integer_split(f2_hi, n2, r2); + + // Add up integer parts and sum the remainders. + Packet n_z = padd(n1, n2); + // Notice: I experimented with using compensated (Kahan) summation here, + // but it does not seem to matter. + Packet rem = padd(padd(f1_lo, f2_lo), padd(r1, r2)); + + // Extract any additional integer part that may have accumulated in rem. + Packet nrem, r_z; + integer_split(rem, nrem, r_z); + n_z = padd(n_z, nrem); + + // We now have an accurate split of f = n_z + r_z and can compute + // x^y = 2**{n_z + r_z) = exp(ln(2) * r_z) * 2**{n_z}. + // The first factor we compute by calling pexp(), while multiplication + // by an integer power of 2 can be done exactly using pldexp(). + // Note: I experimented with using Dekker's algorithms for the + // multiplication by ln(2) here, but did not see any difference. + Packet e_r = pexp(pmul(pset1(Scalar(EIGEN_LN2)), r_z)); + return pldexp(e_r, n_z); +} + +// Generic implementation of pow(x,y). +template +EIGEN_DEFINE_FUNCTION_ALLOWING_MULTIPLE_DEFINITIONS +EIGEN_UNUSED +Packet generic_pow(const Packet& x, const Packet& y) { + typedef typename unpacket_traits::type Scalar; + const Packet cst_pos_inf = pset1(NumTraits::infinity()); + const Packet cst_zero = pset1(Scalar(0)); + const Packet cst_one = pset1(Scalar(1)); + const Packet cst_nan = pset1(NumTraits::quiet_NaN()); + + Packet abs_x = pabs(x); + // Predicates for sign and magnitude of x. + Packet x_is_zero = pcmp_eq(x, cst_zero); + Packet x_is_neg = pcmp_lt(x, cst_zero); + Packet abs_x_is_inf = pcmp_eq(abs_x, cst_pos_inf); + Packet abs_x_is_one = pcmp_eq(abs_x, cst_one); + Packet abs_x_is_gt_one = pcmp_lt(cst_one, abs_x); + Packet abs_x_is_lt_one = pcmp_lt(abs_x, cst_one); + Packet x_is_one = pandnot(abs_x_is_one, x_is_neg); + Packet x_is_neg_one = pand(abs_x_is_one, x_is_neg); + Packet x_is_nan = pandnot(ptrue(x), pcmp_eq(x, x)); + + // Predicates for sign and magnitude of y. + Packet y_is_zero = pcmp_eq(y, cst_zero); + Packet y_is_neg = pcmp_lt(y, cst_zero); + Packet y_is_pos = pandnot(ptrue(y), por(y_is_zero, y_is_neg)); + Packet y_is_nan = pandnot(ptrue(y), pcmp_eq(y, y)); + Packet abs_y_is_inf = pcmp_eq(pabs(y), cst_pos_inf); + + // Predicates for whether y is integer and/or even. + Packet y_is_int = pcmp_eq(pfloor(y), y); + Packet y_div_2 = pldexp(y, pset1(Scalar(-1))); + Packet y_is_even = pcmp_eq(pround(y_div_2), y_div_2); + + // Predicates encoding special cases for the value of pow(x,y) + Packet invalid_negative_x = pandnot(pandnot(pandnot(x_is_neg, abs_x_is_inf), y_is_int), abs_y_is_inf); + Packet pow_is_nan = por(invalid_negative_x, por(x_is_nan, y_is_nan)); + Packet pow_is_one = por(por(y_is_zero, x_is_one), pand(x_is_neg_one, abs_y_is_inf)); + Packet pow_is_zero = por(por(por(pand(x_is_zero, y_is_pos), pand(abs_x_is_inf, y_is_neg)), + pand(pand(abs_x_is_lt_one, abs_y_is_inf), y_is_pos)), + pand(pand(abs_x_is_gt_one, abs_y_is_inf), y_is_neg)); + Packet pow_is_inf = por(por(por(pand(x_is_zero, y_is_neg), pand(abs_x_is_inf, y_is_pos)), + pand(pand(abs_x_is_lt_one, abs_y_is_inf), y_is_neg)), + pand(pand(abs_x_is_gt_one, abs_y_is_inf), y_is_pos)); + + // General computation of pow(x,y) for positive x or negative x and integer y. + Packet negate_pow_abs = pandnot(x_is_neg, y_is_even); + Packet pow_abs = generic_pow_impl(abs_x, y); + + return pselect(pow_is_one, cst_one, + pselect(pow_is_nan, cst_nan, + pselect(pow_is_inf, cst_pos_inf, + pselect(pow_is_zero, cst_zero, + pselect(negate_pow_abs, pnegate(pow_abs), pow_abs))))); +} + + /* polevl (modified for Eigen) * * Evaluate polynomial -- cgit v1.2.3