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//===-- Single-precision general exp/log functions ------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
#define LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
#include "common_constants.h"
#include "src/__support/common.h"
#include "src/__support/macros/properties/cpu_features.h"
#include "src/__support/math/acoshf_utils.h"
#include "src/__support/math/exp10f_utils.h"
#include "src/__support/math/exp_utils.h"
namespace LIBC_NAMESPACE_DECL {
constexpr int LOG_P1_BITS = 6;
constexpr int LOG_P1_SIZE = 1 << LOG_P1_BITS;
// The function correctly calculates sinh(x) and cosh(x) by calculating exp(x)
// and exp(-x) simultaneously.
// To compute e^x, we perform the following range
// reduction: find hi, mid, lo such that:
// x = (hi + mid) * log(2) + lo, in which
// hi is an integer,
// 0 <= mid * 2^5 < 32 is an integer
// -2^(-6) <= lo * log2(e) <= 2^-6.
// In particular,
// hi + mid = round(x * log2(e) * 2^5) * 2^(-5).
// Then,
// e^x = 2^(hi + mid) * e^lo = 2^hi * 2^mid * e^lo.
// 2^mid is stored in the lookup table of 32 elements.
// e^lo is computed using a degree-5 minimax polynomial
// generated by Sollya:
// e^lo ~ P(lo) = 1 + lo + c2 * lo^2 + ... + c5 * lo^5
// = (1 + c2*lo^2 + c4*lo^4) + lo * (1 + c3*lo^2 + c5*lo^4)
// = P_even + lo * P_odd
// We perform 2^hi * 2^mid by simply add hi to the exponent field
// of 2^mid.
// To compute e^(-x), notice that:
// e^(-x) = 2^(-(hi + mid)) * e^(-lo)
// ~ 2^(-(hi + mid)) * P(-lo)
// = 2^(-(hi + mid)) * (P_even - lo * P_odd)
// So:
// sinh(x) = (e^x - e^(-x)) / 2
// ~ 0.5 * (2^(hi + mid) * (P_even + lo * P_odd) -
// 2^(-(hi + mid)) * (P_even - lo * P_odd))
// = 0.5 * (P_even * (2^(hi + mid) - 2^(-(hi + mid))) +
// lo * P_odd * (2^(hi + mid) + 2^(-(hi + mid))))
// And similarly:
// cosh(x) = (e^x + e^(-x)) / 2
// ~ 0.5 * (P_even * (2^(hi + mid) + 2^(-(hi + mid))) +
// lo * P_odd * (2^(hi + mid) - 2^(-(hi + mid))))
// The main point of these formulas is that the expensive part of calculating
// the polynomials approximating lower parts of e^(x) and e^(-x) are shared
// and only done once.
template <bool is_sinh> LIBC_INLINE double exp_pm_eval(float x) {
double xd = static_cast<double>(x);
// kd = round(x * log2(e) * 2^5)
// k_p = round(x * log2(e) * 2^5)
// k_m = round(-x * log2(e) * 2^5)
double kd;
int k_p, k_m;
#ifdef LIBC_TARGET_CPU_HAS_NEAREST_INT
kd = fputil::nearest_integer(ExpBase::LOG2_B * xd);
k_p = static_cast<int>(kd);
k_m = -k_p;
#else
constexpr double HALF_WAY[2] = {0.5, -0.5};
k_p = static_cast<int>(
fputil::multiply_add(xd, ExpBase::LOG2_B, HALF_WAY[x < 0.0f]));
k_m = -k_p;
kd = static_cast<double>(k_p);
#endif // LIBC_TARGET_CPU_HAS_NEAREST_INT
// hi = floor(kf * 2^(-5))
// exp_hi = shift hi to the exponent field of double precision.
int64_t exp_hi_p = static_cast<int64_t>((k_p >> ExpBase::MID_BITS))
<< fputil::FPBits<double>::FRACTION_LEN;
int64_t exp_hi_m = static_cast<int64_t>((k_m >> ExpBase::MID_BITS))
<< fputil::FPBits<double>::FRACTION_LEN;
// mh_p = 2^(hi + mid)
// mh_m = 2^(-(hi + mid))
// mh_bits_* = bit field of mh_*
int64_t mh_bits_p = ExpBase::EXP_2_MID[k_p & ExpBase::MID_MASK] + exp_hi_p;
int64_t mh_bits_m = ExpBase::EXP_2_MID[k_m & ExpBase::MID_MASK] + exp_hi_m;
double mh_p = fputil::FPBits<double>(uint64_t(mh_bits_p)).get_val();
double mh_m = fputil::FPBits<double>(uint64_t(mh_bits_m)).get_val();
// mh_sum = 2^(hi + mid) + 2^(-(hi + mid))
double mh_sum = mh_p + mh_m;
// mh_diff = 2^(hi + mid) - 2^(-(hi + mid))
double mh_diff = mh_p - mh_m;
// dx = lo = x - (hi + mid) * log(2)
double dx =
fputil::multiply_add(kd, ExpBase::M_LOGB_2_LO,
fputil::multiply_add(kd, ExpBase::M_LOGB_2_HI, xd));
double dx2 = dx * dx;
// c0 = 1 + COEFFS[0] * lo^2
// P_even = (1 + COEFFS[0] * lo^2 + COEFFS[2] * lo^4) / 2
double p_even = fputil::polyeval(dx2, 0.5, ExpBase::COEFFS[0] * 0.5,
ExpBase::COEFFS[2] * 0.5);
// P_odd = (1 + COEFFS[1] * lo^2 + COEFFS[3] * lo^4) / 2
double p_odd = fputil::polyeval(dx2, 0.5, ExpBase::COEFFS[1] * 0.5,
ExpBase::COEFFS[3] * 0.5);
double r;
if constexpr (is_sinh)
r = fputil::multiply_add(dx * mh_sum, p_odd, p_even * mh_diff);
else
r = fputil::multiply_add(dx * mh_diff, p_odd, p_even * mh_sum);
return r;
}
// x should be positive, normal finite value
// TODO: Simplify range reduction and polynomial degree for float16.
// See issue #137190.
LIBC_INLINE static float log_eval_f(float x) {
// For x = 2^ex * (1 + mx), logf(x) = ex * logf(2) + logf(1 + mx).
using FPBits = fputil::FPBits<float>;
FPBits xbits(x);
float ex = static_cast<float>(xbits.get_exponent());
// p1 is the leading 7 bits of mx, i.e.
// p1 * 2^(-7) <= m_x < (p1 + 1) * 2^(-7).
int p1 = static_cast<int>(xbits.get_mantissa() >> (FPBits::FRACTION_LEN - 7));
// Set bits to (1 + (mx - p1*2^(-7)))
xbits.set_uintval(xbits.uintval() & (FPBits::FRACTION_MASK >> 7));
xbits.set_biased_exponent(FPBits::EXP_BIAS);
// dx = (mx - p1*2^(-7)) / (1 + p1*2^(-7)).
float dx = (xbits.get_val() - 1.0f) * ONE_OVER_F_FLOAT[p1];
// Minimax polynomial for log(1 + dx), generated using Sollya:
// > P = fpminimax(log(1 + x)/x, 6, [|SG...|], [0, 2^-7]);
// > Q = (P - 1) / x;
// > for i from 0 to degree(Q) do print(coeff(Q, i));
constexpr float COEFFS[6] = {-0x1p-1f, 0x1.555556p-2f, -0x1.00022ep-2f,
0x1.9ea056p-3f, -0x1.e50324p-2f, 0x1.c018fp3f};
float dx2 = dx * dx;
float c1 = fputil::multiply_add(dx, COEFFS[1], COEFFS[0]);
float c2 = fputil::multiply_add(dx, COEFFS[3], COEFFS[2]);
float c3 = fputil::multiply_add(dx, COEFFS[5], COEFFS[4]);
float p = fputil::polyeval(dx2, dx, c1, c2, c3);
// Generated by Sollya with the following commands:
// > display = hexadecimal;
// > round(log(2), SG, RN);
constexpr float LOGF_2 = 0x1.62e43p-1f;
float result = fputil::multiply_add(ex, LOGF_2, LOG_F_FLOAT[p1] + p);
return result;
}
} // namespace LIBC_NAMESPACE_DECL
#endif // LLVM_LIBC_SRC_MATH_GENERIC_EXPLOGXF_H
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