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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 &not)
+ 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
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