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++ // Copyright 2009-2021 Intel Corporation
++ // SPDX-License-Identifier: Apache-2.0
++
++ #pragma once
++
++ // Transcendental functions from "ispc": https://github.com/ispc/ispc/
++ // Most of the transcendental implementations in ispc code come from
++ // Solomon Boulos's "syrah": https://github.com/boulos/syrah/
++
++ #include "../simd/simd.h"
++
++ namespace embree
++ {
++
++ namespace fastapprox
++ {
++
++ template <typename T>
++ __forceinline T sin(const T &v)
++ {
++ static const float piOverTwoVec = 1.57079637050628662109375;
++ static const float twoOverPiVec = 0.636619746685028076171875;
++ auto scaled = v * twoOverPiVec;
++ auto kReal = floor(scaled);
++ auto k = toInt(kReal);
++
++ // Reduced range version of x
++ auto x = v - kReal * piOverTwoVec;
++ auto kMod4 = k & 3;
++ auto sinUseCos = (kMod4 == 1) | (kMod4 == 3);
++ auto flipSign = (kMod4 > 1);
++
++ // These coefficients are from sollya with fpminimax(sin(x)/x, [|0, 2,
++ // 4, 6, 8, 10|], [|single...|], [0;Pi/2]);
++ static const float sinC2 = -0.16666667163372039794921875;
++ static const float sinC4 = +8.333347737789154052734375e-3;
++ static const float sinC6 = -1.9842604524455964565277099609375e-4;
++ static const float sinC8 = +2.760012648650445044040679931640625e-6;
++ static const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
++
++ static const float cosC2 = -0.5;
++ static const float cosC4 = +4.166664183139801025390625e-2;
++ static const float cosC6 = -1.388833043165504932403564453125e-3;
++ static const float cosC8 = +2.47562347794882953166961669921875e-5;
++ static const float cosC10 = -2.59630184018533327616751194000244140625e-7;
++
++ auto outside = select(sinUseCos, 1., x);
++ auto c2 = select(sinUseCos, T(cosC2), T(sinC2));
++ auto c4 = select(sinUseCos, T(cosC4), T(sinC4));
++ auto c6 = select(sinUseCos, T(cosC6), T(sinC6));
++ auto c8 = select(sinUseCos, T(cosC8), T(sinC8));
++ auto c10 = select(sinUseCos, T(cosC10), T(sinC10));
++
++ auto x2 = x * x;
++ auto formula = x2 * c10 + c8;
++ formula = x2 * formula + c6;
++ formula = x2 * formula + c4;
++ formula = x2 * formula + c2;
++ formula = x2 * formula + 1.;
++ formula *= outside;
++
++ formula = select(flipSign, -formula, formula);
++ return formula;
++ }
++
++ template <typename T>
++ __forceinline T cos(const T &v)
++ {
++ static const float piOverTwoVec = 1.57079637050628662109375;
++ static const float twoOverPiVec = 0.636619746685028076171875;
++ auto scaled = v * twoOverPiVec;
++ auto kReal = floor(scaled);
++ auto k = toInt(kReal);
++
++ // Reduced range version of x
++ auto x = v - kReal * piOverTwoVec;
++
++ auto kMod4 = k & 3;
++ auto cosUseCos = (kMod4 == 0) | (kMod4 == 2);
++ auto flipSign = (kMod4 == 1) | (kMod4 == 2);
++
++ const float sinC2 = -0.16666667163372039794921875;
++ const float sinC4 = +8.333347737789154052734375e-3;
++ const float sinC6 = -1.9842604524455964565277099609375e-4;
++ const float sinC8 = +2.760012648650445044040679931640625e-6;
++ const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
++
++ const float cosC2 = -0.5;
++ const float cosC4 = +4.166664183139801025390625e-2;
++ const float cosC6 = -1.388833043165504932403564453125e-3;
++ const float cosC8 = +2.47562347794882953166961669921875e-5;
++ const float cosC10 = -2.59630184018533327616751194000244140625e-7;
++
++ auto outside = select(cosUseCos, 1., x);
++ auto c2 = select(cosUseCos, T(cosC2), T(sinC2));
++ auto c4 = select(cosUseCos, T(cosC4), T(sinC4));
++ auto c6 = select(cosUseCos, T(cosC6), T(sinC6));
++ auto c8 = select(cosUseCos, T(cosC8), T(sinC8));
++ auto c10 = select(cosUseCos, T(cosC10), T(sinC10));
++
++ auto x2 = x * x;
++ auto formula = x2 * c10 + c8;
++ formula = x2 * formula + c6;
++ formula = x2 * formula + c4;
++ formula = x2 * formula + c2;
++ formula = x2 * formula + 1.;
++ formula *= outside;
++
++ formula = select(flipSign, -formula, formula);
++ return formula;
++ }
++
++ template <typename T>
++ __forceinline void sincos(const T &v, T &sinResult, T &cosResult)
++ {
++ const float piOverTwoVec = 1.57079637050628662109375;
++ const float twoOverPiVec = 0.636619746685028076171875;
++ auto scaled = v * twoOverPiVec;
++ auto kReal = floor(scaled);
++ auto k = toInt(kReal);
++
++ // Reduced range version of x
++ auto x = v - kReal * piOverTwoVec;
++ auto kMod4 = k & 3;
++ auto cosUseCos = ((kMod4 == 0) | (kMod4 == 2));
++ auto sinUseCos = ((kMod4 == 1) | (kMod4 == 3));
++ auto sinFlipSign = (kMod4 > 1);
++ auto cosFlipSign = ((kMod4 == 1) | (kMod4 == 2));
++
++ const float oneVec = +1.;
++ const float sinC2 = -0.16666667163372039794921875;
++ const float sinC4 = +8.333347737789154052734375e-3;
++ const float sinC6 = -1.9842604524455964565277099609375e-4;
++ const float sinC8 = +2.760012648650445044040679931640625e-6;
++ const float sinC10 = -2.50293279435709337121807038784027099609375e-8;
++
++ const float cosC2 = -0.5;
++ const float cosC4 = +4.166664183139801025390625e-2;
++ const float cosC6 = -1.388833043165504932403564453125e-3;
++ const float cosC8 = +2.47562347794882953166961669921875e-5;
++ const float cosC10 = -2.59630184018533327616751194000244140625e-7;
++
++ auto x2 = x * x;
++
++ auto sinFormula = x2 * sinC10 + sinC8;
++ auto cosFormula = x2 * cosC10 + cosC8;
++ sinFormula = x2 * sinFormula + sinC6;
++ cosFormula = x2 * cosFormula + cosC6;
++
++ sinFormula = x2 * sinFormula + sinC4;
++ cosFormula = x2 * cosFormula + cosC4;
++
++ sinFormula = x2 * sinFormula + sinC2;
++ cosFormula = x2 * cosFormula + cosC2;
++
++ sinFormula = x2 * sinFormula + oneVec;
++ cosFormula = x2 * cosFormula + oneVec;
++
++ sinFormula *= x;
++
++ sinResult = select(sinUseCos, cosFormula, sinFormula);
++ cosResult = select(cosUseCos, cosFormula, sinFormula);
++
++ sinResult = select(sinFlipSign, -sinResult, sinResult);
++ cosResult = select(cosFlipSign, -cosResult, cosResult);
++ }
++
++ template <typename T>
++ __forceinline T tan(const T &v)
++ {
++ const float piOverFourVec = 0.785398185253143310546875;
++ const float fourOverPiVec = 1.27323949337005615234375;
++
++ auto xLt0 = v < 0.;
++ auto y = select(xLt0, -v, v);
++ auto scaled = y * fourOverPiVec;
++
++ auto kReal = floor(scaled);
++ auto k = toInt(kReal);
++
++ auto x = y - kReal * piOverFourVec;
++
++ // If k & 1, x -= Pi/4
++ auto needOffset = (k & 1) != 0;
++ x = select(needOffset, x - piOverFourVec, x);
++
++ // If k & 3 == (0 or 3) let z = tan_In...(y) otherwise z = -cot_In0To...
++ auto kMod4 = k & 3;
++ auto useCotan = (kMod4 == 1) | (kMod4 == 2);
++
++ const float oneVec = 1.0;
++
++ const float tanC2 = +0.33333075046539306640625;
++ const float tanC4 = +0.13339905440807342529296875;
++ const float tanC6 = +5.3348250687122344970703125e-2;
++ const float tanC8 = +2.46033705770969390869140625e-2;
++ const float tanC10 = +2.892402000725269317626953125e-3;
++ const float tanC12 = +9.500005282461643218994140625e-3;
++
++ const float cotC2 = -0.3333333432674407958984375;
++ const float cotC4 = -2.222204394638538360595703125e-2;
++ const float cotC6 = -2.11752182804048061370849609375e-3;
++ const float cotC8 = -2.0846328698098659515380859375e-4;
++ const float cotC10 = -2.548247357481159269809722900390625e-5;
++ const float cotC12 = -3.5257363606433500535786151885986328125e-7;
++
++ auto x2 = x * x;
++ T z;
++ if (any(useCotan))
++ {
++ auto cotVal = x2 * cotC12 + cotC10;
++ cotVal = x2 * cotVal + cotC8;
++ cotVal = x2 * cotVal + cotC6;
++ cotVal = x2 * cotVal + cotC4;
++ cotVal = x2 * cotVal + cotC2;
++ cotVal = x2 * cotVal + oneVec;
++ // The equation is for x * cot(x) but we need -x * cot(x) for the tan part.
++ cotVal /= -x;
++ z = cotVal;
++ }
++ auto useTan = !useCotan;
++ if (any(useTan))
++ {
++ auto tanVal = x2 * tanC12 + tanC10;
++ tanVal = x2 * tanVal + tanC8;
++ tanVal = x2 * tanVal + tanC6;
++ tanVal = x2 * tanVal + tanC4;
++ tanVal = x2 * tanVal + tanC2;
++ tanVal = x2 * tanVal + oneVec;
++ // Equation was for tan(x)/x
++ tanVal *= x;
++ z = select(useTan, tanVal, z);
++ }
++ return select(xLt0, -z, z);
++ }
++
++ template <typename T>
++ __forceinline T asin(const T &x0)
++ {
++ auto isneg = (x0 < 0.f);
++ auto x = abs(x0);
++ auto isnan = (x > 1.f);
++
++ // sollya
++ // fpminimax(((asin(x)-pi/2)/-sqrt(1-x)), [|0,1,2,3,4,5|],[|single...|],
++ // [1e-20;.9999999999999999]);
++ // avg error: 1.1105439e-06, max error 1.3187528e-06
++ auto v = 1.57079517841339111328125f +
++ x * (-0.21450997889041900634765625f +
++ x * (8.78556668758392333984375e-2f +
++ x * (-4.489909112453460693359375e-2f +
++ x * (1.928029954433441162109375e-2f +
++ x * (-4.3095736764371395111083984375e-3f)))));
++
++ v *= -sqrt(1.f - x);
++ v = v + 1.57079637050628662109375f;
++
++ v = select(v < 0.f, T(0.f), v);
++ v = select(isneg, -v, v);
++ v = select(isnan, T(cast_i2f(0x7fc00000)), v);
++
++ return v;
++ }
++
++ template <typename T>
++ __forceinline T acos(const T &v)
++ {
++ return 1.57079637050628662109375f - asin(v);
++ }
++
++ template <typename T>
++ __forceinline T atan(const T &v)
++ {
++ const float piOverTwoVec = 1.57079637050628662109375;
++ // atan(-x) = -atan(x) (so flip from negative to positive first)
++ // If x > 1 -> atan(x) = Pi/2 - atan(1/x)
++ auto xNeg = v < 0.f;
++ auto xFlipped = select(xNeg, -v, v);
++
++ auto xGt1 = xFlipped > 1.;
++ auto x = select(xGt1, rcpSafe(xFlipped), xFlipped);
++
++ // These coefficients approximate atan(x)/x
++ const float atanC0 = +0.99999988079071044921875;
++ const float atanC2 = -0.3333191573619842529296875;
++ const float atanC4 = +0.199689209461212158203125;
++ const float atanC6 = -0.14015688002109527587890625;
++ const float atanC8 = +9.905083477497100830078125e-2;
++ const float atanC10 = -5.93664981424808502197265625e-2;
++ const float atanC12 = +2.417283318936824798583984375e-2;
++ const float atanC14 = -4.6721356920897960662841796875e-3;
++
++ auto x2 = x * x;
++ auto result = x2 * atanC14 + atanC12;
++ result = x2 * result + atanC10;
++ result = x2 * result + atanC8;
++ result = x2 * result + atanC6;
++ result = x2 * result + atanC4;
++ result = x2 * result + atanC2;
++ result = x2 * result + atanC0;
++ result *= x;
++
++ result = select(xGt1, piOverTwoVec - result, result);
++ result = select(xNeg, -result, result);
++ return result;
++ }
++
++ template <typename T>
++ __forceinline T atan2(const T &y, const T &x)
++ {
++ const float piVec = 3.1415926536;
++ // atan2(y, x) =
++ //
++ // atan2(y > 0, x = +-0) -> Pi/2
++ // atan2(y < 0, x = +-0) -> -Pi/2
++ // atan2(y = +-0, x < +0) -> +-Pi
++ // atan2(y = +-0, x >= +0) -> +-0
++ //
++ // atan2(y >= 0, x < 0) -> Pi + atan(y/x)
++ // atan2(y < 0, x < 0) -> -Pi + atan(y/x)
++ // atan2(y, x > 0) -> atan(y/x)
++ //
++ // and then a bunch of code for dealing with infinities.
++ auto yOverX = y * rcpSafe(x);
++ auto atanArg = atan(yOverX);
++ auto xLt0 = x < 0.f;
++ auto yLt0 = y < 0.f;
++ auto offset = select(xLt0,
++ select(yLt0, T(-piVec), T(piVec)), 0.f);
++ return offset + atanArg;
++ }
++
++ template <typename T>
++ __forceinline T exp(const T &v)
++ {
++ const float ln2Part1 = 0.6931457519;
++ const float ln2Part2 = 1.4286067653e-6;
++ const float oneOverLn2 = 1.44269502162933349609375;
++
++ auto scaled = v * oneOverLn2;
++ auto kReal = floor(scaled);
++ auto k = toInt(kReal);
++
++ // Reduced range version of x
++ auto x = v - kReal * ln2Part1;
++ x -= kReal * ln2Part2;
++
++ // These coefficients are for e^x in [0, ln(2)]
++ const float one = 1.;
++ const float c2 = 0.4999999105930328369140625;
++ const float c3 = 0.166668415069580078125;
++ const float c4 = 4.16539050638675689697265625e-2;
++ const float c5 = 8.378830738365650177001953125e-3;
++ const float c6 = 1.304379315115511417388916015625e-3;
++ const float c7 = 2.7555381529964506626129150390625e-4;
++
++ auto result = x * c7 + c6;
++ result = x * result + c5;
++ result = x * result + c4;
++ result = x * result + c3;
++ result = x * result + c2;
++ result = x * result + one;
++ result = x * result + one;
++
++ // Compute 2^k (should differ for float and double, but I'll avoid
++ // it for now and just do floats)
++ const int fpbias = 127;
++ auto biasedN = k + fpbias;
++ auto overflow = kReal > fpbias;
++ // Minimum exponent is -126, so if k is <= -127 (k + 127 <= 0)
++ // we've got underflow. -127 * ln(2) -> -88.02. So the most
++ // negative float input that doesn't result in zero is like -88.
++ auto underflow = kReal <= -fpbias;
++ const int infBits = 0x7f800000;
++ biasedN <<= 23;
++ // Reinterpret this thing as float
++ auto twoToTheN = asFloat(biasedN);
++ // Handle both doubles and floats (hopefully eliding the copy for float)
++ auto elemtype2n = twoToTheN;
++ result *= elemtype2n;
++ result = select(overflow, cast_i2f(infBits), result);
++ result = select(underflow, 0., result);
++ return result;
++ }
++
++ // Range reduction for logarithms takes log(x) -> log(2^n * y) -> n
++ // * log(2) + log(y) where y is the reduced range (usually in [1/2, 1)).
++ template <typename T, typename R>
++ __forceinline void __rangeReduceLog(const T &input,
++ T &reduced,
++ R &exponent)
++ {
++ auto intVersion = asInt(input);
++ // single precision = SEEE EEEE EMMM MMMM MMMM MMMM MMMM MMMM
++ // exponent mask = 0111 1111 1000 0000 0000 0000 0000 0000
++ // 0x7 0xF 0x8 0x0 0x0 0x0 0x0 0x0
++ // non-exponent = 1000 0000 0111 1111 1111 1111 1111 1111
++ // = 0x8 0x0 0x7 0xF 0xF 0xF 0xF 0xF
++
++ //const int exponentMask(0x7F800000)
++ static const int nonexponentMask = 0x807FFFFF;
++
++ // We want the reduced version to have an exponent of -1 which is
++ // -1 + 127 after biasing or 126
++ static const int exponentNeg1 = (126l << 23);
++ // NOTE(boulos): We don't need to mask anything out since we know
++ // the sign bit has to be 0. If it's 1, we need to return infinity/nan
++ // anyway (log(x), x = +-0 -> infinity, x < 0 -> NaN).
++ auto biasedExponent = intVersion >> 23; // This number is [0, 255] but it means [-127, 128]
++
++ auto offsetExponent = biasedExponent + 1; // Treat the number as if it were 2^{e+1} * (1.m)/2
++ exponent = offsetExponent - 127; // get the real value
++
++ // Blend the offset_exponent with the original input (do this in
++ // int for now, until I decide if float can have & and ¬)
++ auto blended = (intVersion & nonexponentMask) | (exponentNeg1);
++ reduced = asFloat(blended);
++ }
++
++ template <typename T> struct ExponentType { };
++ template <int N> struct ExponentType<vfloat_impl<N>> { typedef vint<N> Ty; };
++ template <> struct ExponentType<float> { typedef int Ty; };
++
++ template <typename T>
++ __forceinline T log(const T &v)
++ {
++ T reduced;
++ typename ExponentType<T>::Ty exponent;
++
++ const int nanBits = 0x7fc00000;
++ const int negInfBits = 0xFF800000;
++ const float nan = cast_i2f(nanBits);
++ const float negInf = cast_i2f(negInfBits);
++ auto useNan = v < 0.;
++ auto useInf = v == 0.;
++ auto exceptional = useNan | useInf;
++ const float one = 1.0;
++
++ auto patched = select(exceptional, one, v);
++ __rangeReduceLog(patched, reduced, exponent);
++
++ const float ln2 = 0.693147182464599609375;
++
++ auto x1 = one - reduced;
++ const float c1 = +0.50000095367431640625;
++ const float c2 = +0.33326041698455810546875;
++ const float c3 = +0.2519190013408660888671875;
++ const float c4 = +0.17541764676570892333984375;
++ const float c5 = +0.3424419462680816650390625;
++ const float c6 = -0.599632322788238525390625;
++ const float c7 = +1.98442304134368896484375;
++ const float c8 = -2.4899270534515380859375;
++ const float c9 = +1.7491014003753662109375;
++
++ auto result = x1 * c9 + c8;
++ result = x1 * result + c7;
++ result = x1 * result + c6;
++ result = x1 * result + c5;
++ result = x1 * result + c4;
++ result = x1 * result + c3;
++ result = x1 * result + c2;
++ result = x1 * result + c1;
++ result = x1 * result + one;
++
++ // Equation was for -(ln(red)/(1-red))
++ result *= -x1;
++ result += toFloat(exponent) * ln2;
++
++ return select(exceptional,
++ select(useNan, T(nan), T(negInf)),
++ result);
++ }
++
++ template <typename T>
++ __forceinline T pow(const T &x, const T &y)
++ {
++ auto x1 = abs(x);
++ auto z = exp(y * log(x1));
++
++ // Handle special cases
++ const float twoOver23 = 8388608.0f;
++ auto yInt = y == round(y);
++ auto yOddInt = select(yInt, asInt(abs(y) + twoOver23) << 31, 0); // set sign bit
++
++ // x == 0
++ z = select(x == 0.0f,
++ select(y < 0.0f, T(inf) | signmsk(x),
++ select(y == 0.0f, T(1.0f), asFloat(yOddInt) & x)), z);
++
++ // x < 0
++ auto xNegative = x < 0.0f;
++ if (any(xNegative))
++ {
++ auto z1 = z | asFloat(yOddInt);
++ z1 = select(yInt, z1, std::numeric_limits<float>::quiet_NaN());
++ z = select(xNegative, z1, z);
++ }
++
++ auto xFinite = isfinite(x);
++ auto yFinite = isfinite(y);
++ if (all(xFinite & yFinite))
++ return z;
++
++ // x finite and y infinite
++ z = select(andn(xFinite, yFinite),
++ select(x1 == 1.0f, 1.0f,
++ select((x1 > 1.0f) ^ (y < 0.0f), inf, T(0.0f))), z);
++
++ // x infinite
++ z = select(xFinite, z,
++ select(y == 0.0f, 1.0f,
++ select(y < 0.0f, T(0.0f), inf) | (asFloat(yOddInt) & x)));
++
++ return z;
++ }
++
++ template <typename T>
++ __forceinline T pow(const T &x, float y)
++ {
++ return pow(x, T(y));
++ }
++
++ } // namespace fastapprox
++
++ } // namespace embree