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// Copyright (c) the JPEG XL Project Authors. All rights reserved.
//
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Fast SIMD math ops (log2, encoder only, cos, erf for splines)
#if defined(LIB_JXL_BASE_FAST_MATH_INL_H_) == defined(HWY_TARGET_TOGGLE)
#ifdef LIB_JXL_BASE_FAST_MATH_INL_H_
#undef LIB_JXL_BASE_FAST_MATH_INL_H_
#else
#define LIB_JXL_BASE_FAST_MATH_INL_H_
#endif
#include <hwy/highway.h>
#include "lib/jxl/base/common.h"
#include "lib/jxl/base/rational_polynomial-inl.h"
HWY_BEFORE_NAMESPACE();
namespace jxl {
namespace HWY_NAMESPACE {
// These templates are not found via ADL.
using hwy::HWY_NAMESPACE::Abs;
using hwy::HWY_NAMESPACE::Add;
using hwy::HWY_NAMESPACE::Eq;
using hwy::HWY_NAMESPACE::Floor;
using hwy::HWY_NAMESPACE::Ge;
using hwy::HWY_NAMESPACE::GetLane;
using hwy::HWY_NAMESPACE::IfThenElse;
using hwy::HWY_NAMESPACE::IfThenZeroElse;
using hwy::HWY_NAMESPACE::Le;
using hwy::HWY_NAMESPACE::Min;
using hwy::HWY_NAMESPACE::Mul;
using hwy::HWY_NAMESPACE::MulAdd;
using hwy::HWY_NAMESPACE::NegMulAdd;
using hwy::HWY_NAMESPACE::Rebind;
using hwy::HWY_NAMESPACE::ShiftLeft;
using hwy::HWY_NAMESPACE::ShiftRight;
using hwy::HWY_NAMESPACE::Sub;
using hwy::HWY_NAMESPACE::Xor;
// Computes base-2 logarithm like std::log2. Undefined if negative / NaN.
// L1 error ~3.9E-6
template <class DF, class V>
V FastLog2f(const DF df, V x) {
// 2,2 rational polynomial approximation of std::log1p(x) / std::log(2).
HWY_ALIGN const float p[4 * (2 + 1)] = {HWY_REP4(-1.8503833400518310E-06f),
HWY_REP4(1.4287160470083755E+00f),
HWY_REP4(7.4245873327820566E-01f)};
HWY_ALIGN const float q[4 * (2 + 1)] = {HWY_REP4(9.9032814277590719E-01f),
HWY_REP4(1.0096718572241148E+00f),
HWY_REP4(1.7409343003366853E-01f)};
const Rebind<int32_t, DF> di;
const auto x_bits = BitCast(di, x);
// Range reduction to [-1/3, 1/3] - 3 integer, 2 float ops
const auto exp_bits = Sub(x_bits, Set(di, 0x3f2aaaab)); // = 2/3
// Shifted exponent = log2; also used to clear mantissa.
const auto exp_shifted = ShiftRight<23>(exp_bits);
const auto mantissa = BitCast(df, Sub(x_bits, ShiftLeft<23>(exp_shifted)));
const auto exp_val = ConvertTo(df, exp_shifted);
return Add(EvalRationalPolynomial(df, Sub(mantissa, Set(df, 1.0f)), p, q),
exp_val);
}
// max relative error ~3e-7
template <class DF, class V>
V FastPow2f(const DF df, V x) {
const Rebind<int32_t, DF> di;
auto floorx = Floor(x);
auto exp =
BitCast(df, ShiftLeft<23>(Add(ConvertTo(di, floorx), Set(di, 127))));
auto frac = Sub(x, floorx);
auto num = Add(frac, Set(df, 1.01749063e+01));
num = MulAdd(num, frac, Set(df, 4.88687798e+01));
num = MulAdd(num, frac, Set(df, 9.85506591e+01));
num = Mul(num, exp);
auto den = MulAdd(frac, Set(df, 2.10242958e-01), Set(df, -2.22328856e-02));
den = MulAdd(den, frac, Set(df, -1.94414990e+01));
den = MulAdd(den, frac, Set(df, 9.85506633e+01));
return Div(num, den);
}
// max relative error ~3e-5
template <class DF, class V>
V FastPowf(const DF df, V base, V exponent) {
return FastPow2f(df, Mul(FastLog2f(df, base), exponent));
}
// Computes cosine like std::cos.
// L1 error 7e-5.
template <class DF, class V>
V FastCosf(const DF df, V x) {
// Step 1: range reduction to [0, 2pi)
const auto pi2 = Set(df, kPi * 2.0f);
const auto pi2_inv = Set(df, 0.5f / kPi);
const auto npi2 = Mul(Floor(Mul(x, pi2_inv)), pi2);
const auto xmodpi2 = Sub(x, npi2);
// Step 2: range reduction to [0, pi]
const auto x_pi = Min(xmodpi2, Sub(pi2, xmodpi2));
// Step 3: range reduction to [0, pi/2]
const auto above_pihalf = Ge(x_pi, Set(df, kPi / 2.0f));
const auto x_pihalf = IfThenElse(above_pihalf, Sub(Set(df, kPi), x_pi), x_pi);
// Step 4: Taylor-like approximation, scaled by 2**0.75 to make angle
// duplication steps faster, on x/4.
const auto xs = Mul(x_pihalf, Set(df, 0.25f));
const auto x2 = Mul(xs, xs);
const auto x4 = Mul(x2, x2);
const auto cosx_prescaling =
MulAdd(x4, Set(df, 0.06960438),
MulAdd(x2, Set(df, -0.84087373), Set(df, 1.68179268)));
// Step 5: angle duplication.
const auto cosx_scale1 =
MulAdd(cosx_prescaling, cosx_prescaling, Set(df, -1.414213562));
const auto cosx_scale2 = MulAdd(cosx_scale1, cosx_scale1, Set(df, -1));
// Step 6: change sign if needed.
const Rebind<uint32_t, DF> du;
auto signbit = ShiftLeft<31>(BitCast(du, VecFromMask(df, above_pihalf)));
return BitCast(df, Xor(signbit, BitCast(du, cosx_scale2)));
}
// Computes the error function like std::erf.
// L1 error 7e-4.
template <class DF, class V>
V FastErff(const DF df, V x) {
// Formula from
// but constants have been recomputed.
const auto xle0 = Le(x, Zero(df));
const auto absx = Abs(x);
// Compute 1 - 1 / ((((x * a + b) * x + c) * x + d) * x + 1)**4
const auto denom1 =
MulAdd(absx, Set(df, 7.77394369e-02), Set(df, 2.05260015e-04));
const auto denom2 = MulAdd(denom1, absx, Set(df, 2.32120216e-01));
const auto denom3 = MulAdd(denom2, absx, Set(df, 2.77820801e-01));
const auto denom4 = MulAdd(denom3, absx, Set(df, 1.0f));
const auto denom5 = Mul(denom4, denom4);
const auto inv_denom5 = Div(Set(df, 1.0f), denom5);
const auto result = NegMulAdd(inv_denom5, inv_denom5, Set(df, 1.0f));
// Change sign if needed.
const Rebind<uint32_t, DF> du;
auto signbit = ShiftLeft<31>(BitCast(du, VecFromMask(df, xle0)));
return BitCast(df, Xor(signbit, BitCast(du, result)));
}
inline float FastLog2f(float f) {
HWY_CAPPED(float, 1) D;
return GetLane(FastLog2f(D, Set(D, f)));
}
inline float FastPow2f(float f) {
HWY_CAPPED(float, 1) D;
return GetLane(FastPow2f(D, Set(D, f)));
}
inline float FastPowf(float b, float e) {
HWY_CAPPED(float, 1) D;
return GetLane(FastPowf(D, Set(D, b), Set(D, e)));
}
inline float FastCosf(float f) {
HWY_CAPPED(float, 1) D;
return GetLane(FastCosf(D, Set(D, f)));
}
inline float FastErff(float f) {
HWY_CAPPED(float, 1) D;
return GetLane(FastErff(D, Set(D, f)));
}
// Returns cbrt(x) + add with 6 ulp max error.
// Modified from vectormath_exp.h, Apache 2 license.
template <class V>
V CubeRootAndAdd(const V x, const V add) {
const HWY_FULL(float) df;
const HWY_FULL(int32_t) di;
const auto kExpBias = Set(di, 0x54800000); // cast(1.) + cast(1.) / 3
const auto kExpMul = Set(di, 0x002AAAAA); // shifted 1/3
const auto k1_3 = Set(df, 1.0f / 3);
const auto k4_3 = Set(df, 4.0f / 3);
const auto xa = x; // assume inputs never negative
const auto xa_3 = Mul(k1_3, xa);
// Multiply exponent by -1/3
const auto m1 = BitCast(di, xa);
// Special case for 0. 0 is represented with an exponent of 0, so the
// "kExpBias - 1/3 * exp" below gives the wrong result. The IfThenZeroElse()
// sets those values as 0, which prevents having NaNs in the computations
// below.
// TODO(eustas): use fused op
const auto m2 = IfThenZeroElse(
Eq(m1, Zero(di)), Sub(kExpBias, Mul((ShiftRight<23>(m1)), kExpMul)));
auto r = BitCast(df, m2);
// Newton-Raphson iterations
for (int i = 0; i < 3; i++) {
const auto r2 = Mul(r, r);
r = NegMulAdd(xa_3, Mul(r2, r2), Mul(k4_3, r));
}
// Final iteration
auto r2 = Mul(r, r);
r = MulAdd(k1_3, NegMulAdd(xa, Mul(r2, r2), r), r);
r2 = Mul(r, r);
r = MulAdd(r2, x, add);
return r;
}
// NOLINTNEXTLINE(google-readability-namespace-comments)
} // namespace HWY_NAMESPACE
} // namespace jxl
HWY_AFTER_NAMESPACE();
#endif // LIB_JXL_BASE_FAST_MATH_INL_H_
#if HWY_ONCE
#ifndef LIB_JXL_BASE_FAST_MATH_ONCE
#define LIB_JXL_BASE_FAST_MATH_ONCE
namespace jxl {
inline float FastLog2f(float f) { return HWY_STATIC_DISPATCH(FastLog2f)(f); }
inline float FastPow2f(float f) { return HWY_STATIC_DISPATCH(FastPow2f)(f); }
inline float FastPowf(float b, float e) {
return HWY_STATIC_DISPATCH(FastPowf)(b, e);
}
inline float FastCosf(float f) { return HWY_STATIC_DISPATCH(FastCosf)(f); }
inline float FastErff(float f) { return HWY_STATIC_DISPATCH(FastErff)(f); }
} // namespace jxl
#endif // LIB_JXL_BASE_FAST_MATH_ONCE
#endif // HWY_ONCE