/*
* Copyright (c) 2021, Leon Albrecht <leon2002.la@gmail.com>.
*
* SPDX-License-Identifier: BSD-2-Clause
*/
#pragma once
#include <AK/BuiltinWrappers.h>
#include <AK/Concepts.h>
#include <AK/FloatingPoint.h>
#include <AK/NumericLimits.h>
#include <AK/StdLibExtraDetails.h>
#include <AK/Types.h>
#include <math.h>
#ifdef KERNEL
# error "Including AK/Math.h from the Kernel is never correct! Floating point is disabled."
#endif
namespace AK {
template<FloatingPoint T>
constexpr T NaN = __builtin_nan("");
template<FloatingPoint T>
constexpr T Infinity = __builtin_huge_vall();
template<FloatingPoint T>
constexpr T Pi = 3.141592653589793238462643383279502884L;
template<FloatingPoint T>
constexpr T E = 2.718281828459045235360287471352662498L;
template<FloatingPoint T>
constexpr T Sqrt2 = 1.414213562373095048801688724209698079L;
template<FloatingPoint T>
constexpr T Sqrt1_2 = 0.707106781186547524400844362104849039L;
template<FloatingPoint T>
constexpr T L2_10 = 3.321928094887362347870319429489390175864L;
template<FloatingPoint T>
constexpr T L2_E = 1.442695040888963407359924681001892137L;
namespace Details {
template<size_t>
constexpr size_t product_even();
template<>
constexpr size_t product_even<2>() { return 2; }
template<size_t value>
constexpr size_t product_even() { return value * product_even<value - 2>(); }
template<size_t>
constexpr size_t product_odd();
template<>
constexpr size_t product_odd<1>() { return 1; }
template<size_t value>
constexpr size_t product_odd() { return value * product_odd<value - 2>(); }
}
template<FloatingPoint T>
constexpr T to_radians(T degrees)
{
return degrees * AK::Pi<T> / 180;
}
template<FloatingPoint T>
constexpr T to_degrees(T radians)
{
return radians * 180 / AK::Pi<T>;
}
template<FloatingPoint FloatT>
FloatT copysign(FloatT x, FloatT y)
{
using Extractor = FloatExtractor<FloatT>;
auto ex = Extractor::from_float(x);
auto ey = Extractor::from_float(y);
ex.sign = ey.sign;
return ex.to_float();
}
#define CONSTEXPR_STATE(function, args...) \
if (is_constant_evaluated()) { \
if (IsSame<T, long double>) \
return __builtin_##function##l(args); \
if (IsSame<T, double>) \
return __builtin_##function(args); \
if (IsSame<T, float>) \
return __builtin_##function##f(args); \
}
#define AARCH64_INSTRUCTION(instruction, arg) \
if constexpr (IsSame<T, long double>) \
TODO(); \
if constexpr (IsSame<T, double>) { \
double res; \
asm(#instruction " %d0, %d1" \
: "=w"(res) \
: "w"(arg)); \
return res; \
} \
if constexpr (IsSame<T, float>) { \
float res; \
asm(#instruction " %s0, %s1" \
: "=w"(res) \
: "w"(arg)); \
return res; \
}
template<FloatingPoint T>
constexpr T fabs(T x)
{
// Both GCC and Clang inline fabs by default, so this is just a cmath like wrapper
if constexpr (IsSame<T, long double>)
return __builtin_fabsl(x);
if constexpr (IsSame<T, double>)
return __builtin_fabs(x);
if constexpr (IsSame<T, float>)
return __builtin_fabsf(x);
}
namespace Rounding {
template<FloatingPoint T>
constexpr T ceil(T num)
{
// FIXME: SSE4.1 rounds[sd] num, res, 0b110
if (is_constant_evaluated()) {
if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
return num;
return (static_cast<T>(static_cast<i64>(num)) == num)
? static_cast<i64>(num)
: static_cast<i64>(num) + ((num > 0) ? 1 : 0);
}
#if ARCH(AARCH64)
AARCH64_INSTRUCTION(frintp, num);
#else
if constexpr (IsSame<T, long double>)
return __builtin_ceill(num);
if constexpr (IsSame<T, double>)
return __builtin_ceil(num);
if constexpr (IsSame<T, float>)
return __builtin_ceilf(num);
#endif
}
template<FloatingPoint T>
constexpr T floor(T num)
{
// FIXME: SSE4.1 rounds[sd] num, res, 0b101
if (is_constant_evaluated()) {
if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
return num;
return (static_cast<T>(static_cast<i64>(num)) == num)
? static_cast<i64>(num)
: static_cast<i64>(num) - ((num > 0) ? 0 : 1);
}
#if ARCH(AARCH64)
AARCH64_INSTRUCTION(frintm, num);
#else
if constexpr (IsSame<T, long double>)
return __builtin_floorl(num);
if constexpr (IsSame<T, double>)
return __builtin_floor(num);
if constexpr (IsSame<T, float>)
return __builtin_floorf(num);
#endif
}
template<FloatingPoint T>
constexpr T trunc(T num)
{
#if ARCH(AARCH64)
if (is_constant_evaluated()) {
if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
return num;
return static_cast<T>(static_cast<i64>(num));
}
AARCH64_INSTRUCTION(frintz, num);
#endif
// FIXME: Use dedicated instruction in the non constexpr case
// SSE4.1: rounds[sd] %num, %res, 0b111
if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
return num;
return static_cast<T>(static_cast<i64>(num));
}
template<FloatingPoint T>
constexpr T rint(T x)
{
CONSTEXPR_STATE(rint, x);
// Note: This does break tie to even
// But the behavior of frndint/rounds[ds]/frintx can be configured
// through the floating point control registers.
// FIXME: We should decide if we rename this to allow us to get away from
// the configurability "burden" rint has
// this would make us use `rounds[sd] %num, %res, 0b100`
// and `frintn` respectively,
// no such guaranteed round exists for x87 `frndint`
#if ARCH(X86_64)
# ifdef __SSE4_1__
if constexpr (IsSame<T, double>) {
T r;
asm(
"roundsd %1, %0"
: "=x"(r)
: "x"(x));
return r;
}
if constexpr (IsSame<T, float>) {
T r;
asm(
"roundss %1, %0"
: "=x"(r)
: "x"(x));
return r;
}
# else
asm(
"frndint"
: "+t"(x));
return x;
# endif
#elif ARCH(AARCH64)
AARCH64_INSTRUCTION(frintx, x);
#elif ARCH(RISCV64)
if (__builtin_isnan(x))
return x;
// Floating point values have a gap size of >= 1 for values above 2^mantissa_bits - 1.
if (fabs(x) > FloatExtractor<T>::mantissa_max)
return x;
if constexpr (IsSame<T, float>) {
i64 r;
asm("fcvt.l.s %0, %1, dyn"
: "=r"(r)
: "f"(x));
return copysign(static_cast<float>(r), x);
}
if constexpr (IsSame<T, double>) {
i64 r;
asm("fcvt.l.d %0, %1, dyn"
: "=r"(r)
: "f"(x));
return copysign(static_cast<double>(r), x);
}
if constexpr (IsSame<T, long double>)
TODO_RISCV64();
#endif
TODO();
}
template<FloatingPoint T>
constexpr T round(T x)
{
CONSTEXPR_STATE(round, x);
// Note: This is break-tie-away-from-zero, so not the hw's understanding of
// "nearest", which would be towards even.
if (x == 0.)
return x;
if (x > 0.)
return floor(x + .5);
return ceil(x - .5);
}
template<Integral I, FloatingPoint P>
ALWAYS_INLINE I round_to(P value);
#if ARCH(X86_64)
template<Integral I>
ALWAYS_INLINE I round_to(long double value)
{
// Note: fistps outputs into a signed integer location (i16, i32, i64),
// so lets be nice and tell the compiler that.
Conditional<sizeof(I) >= sizeof(i16), MakeSigned<I>, i16> ret;
if constexpr (sizeof(I) == sizeof(i64)) {
asm("fistpll %0"
: "=m"(ret)
: "t"(value)
: "st");
} else if constexpr (sizeof(I) == sizeof(i32)) {
asm("fistpl %0"
: "=m"(ret)
: "t"(value)
: "st");
} else {
asm("fistps %0"
: "=m"(ret)
: "t"(value)
: "st");
}
return static_cast<I>(ret);
}
template<Integral I>
ALWAYS_INLINE I round_to(float value)
{
// FIXME: round_to<u64> might will cause issues, aka the indefinite value being set,
// if the value surpasses the i64 limit, even if the result could fit into an u64
// To solve this we would either need to detect that value or do a range check and
// then do a more specialized conversion, which might include a division (which is expensive)
if constexpr (sizeof(I) == sizeof(i64) || IsSame<I, u32>) {
i64 ret;
asm("cvtss2si %1, %0"
: "=r"(ret)
: "xm"(value));
return static_cast<I>(ret);
}
i32 ret;
asm("cvtss2si %1, %0"
: "=r"(ret)
: "xm"(value));
return static_cast<I>(ret);
}
template<Integral I>
ALWAYS_INLINE I round_to(double value)
{
// FIXME: round_to<u64> might will cause issues, aka the indefinite value being set,
// if the value surpasses the i64 limit, even if the result could fit into an u64
// To solve this we would either need to detect that value or do a range check and
// then do a more specialized conversion, which might include a division (which is expensive)
if constexpr (sizeof(I) == sizeof(i64) || IsSame<I, u32>) {
i64 ret;
asm("cvtsd2si %1, %0"
: "=r"(ret)
: "xm"(value));
return static_cast<I>(ret);
}
i32 ret;
asm("cvtsd2si %1, %0"
: "=r"(ret)
: "xm"(value));
return static_cast<I>(ret);
}
#elif ARCH(AARCH64)
template<Signed I>
ALWAYS_INLINE I round_to(float value)
{
if constexpr (sizeof(I) <= sizeof(u32)) {
i32 res;
asm("fcvtns %w0, %s1"
: "=r"(res)
: "w"(value));
return static_cast<I>(res);
}
i64 res;
asm("fcvtns %0, %s1"
: "=r"(res)
: "w"(value));
return static_cast<I>(res);
}
template<Signed I>
ALWAYS_INLINE I round_to(double value)
{
if constexpr (sizeof(I) <= sizeof(u32)) {
i32 res;
asm("fcvtns %w0, %d1"
: "=r"(res)
: "w"(value));
return static_cast<I>(res);
}
i64 res;
asm("fcvtns %0, %d1"
: "=r"(res)
: "w"(value));
return static_cast<I>(res);
}
template<Unsigned U>
ALWAYS_INLINE U round_to(float value)
{
if constexpr (sizeof(U) <= sizeof(u32)) {
u32 res;
asm("fcvtnu %w0, %s1"
: "=r"(res)
: "w"(value));
return static_cast<U>(res);
}
i64 res;
asm("fcvtnu %0, %s1"
: "=r"(res)
: "w"(value));
return static_cast<U>(res);
}
template<Unsigned U>
ALWAYS_INLINE U round_to(double value)
{
if constexpr (sizeof(U) <= sizeof(u32)) {
u32 res;
asm("fcvtns %w0, %d1"
: "=r"(res)
: "w"(value));
return static_cast<U>(res);
}
i64 res;
asm("fcvtns %0, %d1"
: "=r"(res)
: "w"(value));
return static_cast<U>(res);
}
#else
template<Integral I, FloatingPoint P>
ALWAYS_INLINE I round_to(P value)
{
if constexpr (IsSame<P, long double>)
return static_cast<I>(__builtin_llrintl(value));
if constexpr (IsSame<P, double>)
return static_cast<I>(__builtin_llrint(value));
if constexpr (IsSame<P, float>)
return static_cast<I>(__builtin_llrintf(value));
}
#endif
}
using Rounding::ceil;
using Rounding::floor;
using Rounding::rint;
using Rounding::round;
using Rounding::round_to;
using Rounding::trunc;
namespace Division {
template<FloatingPoint T>
constexpr T fmod(T x, T y)
{
CONSTEXPR_STATE(fmod, x, y);
#if ARCH(X86_64)
u16 fpu_status;
do {
asm(
"fprem\n"
"fnstsw %%ax\n"
: "+t"(x), "=a"(fpu_status)
: "u"(y));
} while (fpu_status & 0x400);
return x;
#else
# if defined(AK_OS_MINERVA)
// FIXME: This is a very naive implementation.
if (__builtin_isnan(x))
return x;
if (__builtin_isnan(y))
return y;
// SPECIAL VALUES
// fmod(±0, y) returns ±0 if y is neither 0 nor NaN.
if (x == 0 && y != 0)
return x;
// fmod(x, y) returns a NaN and raises the "invalid" floating-point exception for x infinite or y zero.
// FIXME: Exception.
if (__builtin_isinf(x) || y == 0)
return NaN<T>;
// fmod(x, ±infinity) returns x for x not infinite.
if (__builtin_isinf(y))
return x;
// Range reduction: handle negative x and y.
if (y < 0)
return fmod(x, -y);
if (x < 0)
return -fmod(-x, y);
// x is x_mantissa * 2**x_exponent, y is y_mantissa * 2**y_exponent.
// (Both are positive at this point.)
// If y_exponent > x_exponent, we are done and can return x.
// If y_exponent == x_exponent, we can return (x_mantissa % y_mantissa) * 2**x_exponent.
// If y_exponent < x_exponent, we'll iteratively reduce x_exponent by shifting from
// the exponent into the mantissa.
auto x_bits = FloatExtractor<T>::from_float(x);
typename FloatExtractor<T>::ComponentType x_exponent = x_bits.exponent;
auto y_bits = FloatExtractor<T>::from_float(y);
typename FloatExtractor<T>::ComponentType y_exponent = y_bits.exponent;
// FIXME: Handle denormals. For now, treat them as 0.
if (x_exponent == 0 && y_exponent != 0)
return 0;
if (y_exponent == 0)
return NaN<T>;
if (y_exponent > x_exponent)
return x;
// FIXME: This is wrong for f80.
typename FloatExtractor<T>::ComponentType implicit_mantissa_bit = 1;
implicit_mantissa_bit <<= FloatExtractor<T>::mantissa_bits;
typename FloatExtractor<T>::ComponentType x_mantissa = x_bits.mantissa | implicit_mantissa_bit;
typename FloatExtractor<T>::ComponentType y_mantissa = y_bits.mantissa | implicit_mantissa_bit;
while (y_exponent < x_exponent) {
// This is ok because (x % (y * 2**n)) divides (x % y) for all n > 0.
x_mantissa %= y_mantissa;
x_mantissa <<= 1;
--x_exponent;
}
x_mantissa %= y_mantissa;
// We're done and want to return x_mantissa * 2 ** x_exponent.
// But x_mantissa might not have a leading 1 bit, so we have to realign first.
// Mantissa is mantissa_bits long, count_leading_zeroes() counts in ComponentType, adjust:
auto const always_zero_bits = sizeof(typename FloatExtractor<T>::ComponentType) * 8 - (FloatExtractor<T>::mantissa_bits + 1); // +1 for implicit 1 bit
auto shift = count_leading_zeroes(x_mantissa) - always_zero_bits;
if (x_exponent < shift) {
// FIXME: Make a real denormal.
return 0;
}
x_mantissa <<= shift;
x_exponent -= shift;
x_bits.exponent = x_exponent;
x_bits.mantissa = x_mantissa;
return x_bits.to_float();
# else
if constexpr (IsSame<T, long double>)
return __builtin_fmodl(x, y);
if constexpr (IsSame<T, double>)
return __builtin_fmod(x, y);
if constexpr (IsSame<T, float>)
return __builtin_fmodf(x, y);
# endif
#endif
}
template<FloatingPoint T>
constexpr T remainder(T x, T y)
{
CONSTEXPR_STATE(remainder, x, y);
#if ARCH(X86_64)
u16 fpu_status;
do {
asm(
"fprem1\n"
"fnstsw %%ax\n"
: "+t"(x), "=a"(fpu_status)
: "u"(y));
} while (fpu_status & 0x400);
return x;
#else
# if defined(AK_OS_MINERVA)
// TODO: Add implementation for this function.
TODO();
# endif
if constexpr (IsSame<T, long double>)
return __builtin_remainderl(x, y);
if constexpr (IsSame<T, double>)
return __builtin_remainder(x, y);
if constexpr (IsSame<T, float>)
return __builtin_remainderf(x, y);
#endif
}
}
using Division::fmod;
using Division::remainder;
template<FloatingPoint T>
constexpr T sqrt(T x)
{
CONSTEXPR_STATE(sqrt, x);
#if ARCH(X86_64)
if constexpr (IsSame<T, float>) {
float res;
asm("sqrtss %1, %0"
: "=x"(res)
: "x"(x));
return res;
}
if constexpr (IsSame<T, double>) {
double res;
asm("sqrtsd %1, %0"
: "=x"(res)
: "x"(x));
return res;
}
T res;
asm("fsqrt"
: "=t"(res)
: "0"(x));
return res;
#elif ARCH(AARCH64)
AARCH64_INSTRUCTION(fsqrt, x);
#elif ARCH(RISCV64)
if constexpr (IsSame<T, float>) {
float res;
asm("fsqrt.s %0, %1"
: "=f"(res)
: "f"(x));
return res;
}
if constexpr (IsSame<T, double>) {
double res;
asm("fsqrt.d %0, %1"
: "=f"(res)
: "f"(x));
return res;
}
if constexpr (IsSame<T, long double>)
TODO_RISCV64();
#else
# if defined(AK_OS_MINERVA)
// TODO: Add implementation for this function.
TODO();
# endif
return __builtin_sqrt(x);
#endif
}
template<FloatingPoint T>
constexpr T cbrt(T x)
{
CONSTEXPR_STATE(cbrt, x);
if (__builtin_isinf(x) || x == 0)
return x;
if (x < 0)
return -cbrt(-x);
T r = x;
T ex = 0;
while (r < 0.125l) {
r *= 8;
ex--;
}
while (r > 1.0l) {
r *= 0.125l;
ex++;
}
r = (-0.46946116l * r + 1.072302l) * r + 0.3812513l;
while (ex < 0) {
r *= 0.5l;
ex++;
}
while (ex > 0) {
r *= 2.0l;
ex--;
}
r = (2.0l / 3.0l) * r + (1.0l / 3.0l) * x / (r * r);
r = (2.0l / 3.0l) * r + (1.0l / 3.0l) * x / (r * r);
r = (2.0l / 3.0l) * r + (1.0l / 3.0l) * x / (r * r);
r = (2.0l / 3.0l) * r + (1.0l / 3.0l) * x / (r * r);
return r;
}
namespace Trigonometry {
template<FloatingPoint T>
constexpr T hypot(T x, T y)
{
return sqrt(x * x + y * y);
}
template<FloatingPoint T>
constexpr T sin(T angle)
{
CONSTEXPR_STATE(sin, angle);
#if ARCH(X86_64)
T ret;
asm(
"fsin"
: "=t"(ret)
: "0"(angle));
return ret;
#else
# if defined(AK_OS_MINERVA)
if (angle < 0)
return -sin(-angle);
angle = fmod(angle, 2 * Pi<T>);
if (angle >= Pi<T>)
return -sin(angle - Pi<T>);
if (angle > Pi<T> / 2)
return sin(Pi<T> - angle);
// https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_approximation_formula
// FIXME: This is not a good formula! It requires divisions, so it's slow, and it's not very accurate either.
return 16 * angle * (Pi<T> - angle) / (5 * Pi<T> * Pi<T> - 4 * angle * (Pi<T> - angle));
# else
return __builtin_sin(angle);
# endif
#endif
}
template<FloatingPoint T>
constexpr T cos(T angle)
{
CONSTEXPR_STATE(cos, angle);
#if ARCH(X86_64)
T ret;
asm(
"fcos"
: "=t"(ret)
: "0"(angle));
return ret;
#else
# if defined(AK_OS_MINERVA)
if (angle < 0)
return cos(-angle);
angle = fmod(angle, 2 * Pi<T>);
if (angle >= Pi<T>)
return -cos(angle - Pi<T>);
if (angle > Pi<T> / 2)
return -cos(Pi<T> - angle);
// https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_approximation_formula
// FIXME: This is not a good formula! It requires divisions, so it's slow, and it's not very accurate either.
return (Pi<T> * Pi<T> - 4 * angle * angle) / (Pi<T> * Pi<T> + angle * angle);
# else
return __builtin_cos(angle);
# endif
#endif
}
template<FloatingPoint T>
constexpr void sincos(T angle, T& sin_val, T& cos_val)
{
if (is_constant_evaluated()) {
sin_val = sin(angle);
cos_val = cos(angle);
return;
}
#if ARCH(X86_64)
asm(
"fsincos"
: "=t"(cos_val), "=u"(sin_val)
: "0"(angle));
#else
sin_val = sin(angle);
cos_val = cos(angle);
#endif
}
template<FloatingPoint T>
constexpr T tan(T angle)
{
CONSTEXPR_STATE(tan, angle);
#if ARCH(X86_64)
T ret, one;
asm(
"fptan"
: "=t"(one), "=u"(ret)
: "0"(angle));
return ret;
#else
# if defined(AK_OS_MINERVA)
return sin(angle) / cos(angle);
# else
return __builtin_tan(angle);
# endif
#endif
}
template<FloatingPoint T>
constexpr T asin(T x)
{
CONSTEXPR_STATE(asin, x);
if (x < 0)
return -asin(-x);
if (x > 1)
return NaN<T>;
if (x > (T)0.5) {
// asin(x) = pi/2 - 2 * asin(sqrt((1 - x) / 2))
// If 0.5 < x <= 1, then sqrt((1 - x) / 2 ) < 0.5,
// and the recursion will go down the `<= 0.5` branch.
return Pi<T> / 2 - 2 * asin(sqrt((1 - x) / 2));
}
T squared = x * x;
T value = x;
T i = x * squared;
value += i * Details::product_odd<1>() / Details::product_even<2>() / 3;
i *= squared;
value += i * Details::product_odd<3>() / Details::product_even<4>() / 5;
i *= squared;
value += i * Details::product_odd<5>() / Details::product_even<6>() / 7;
i *= squared;
value += i * Details::product_odd<7>() / Details::product_even<8>() / 9;
i *= squared;
value += i * Details::product_odd<9>() / Details::product_even<10>() / 11;
i *= squared;
value += i * Details::product_odd<11>() / Details::product_even<12>() / 13;
i *= squared;
value += i * Details::product_odd<13>() / Details::product_even<14>() / 15;
i *= squared;
value += i * Details::product_odd<15>() / Details::product_even<16>() / 17;
return value;
}
template<FloatingPoint T>
constexpr T acos(T value)
{
CONSTEXPR_STATE(acos, value);
// FIXME: I am naive
return static_cast<T>(0.5) * Pi<T> - asin<T>(value);
}
template<FloatingPoint T>
constexpr T atan(T value)
{
CONSTEXPR_STATE(atan, value);
#if ARCH(X86_64)
T ret;
asm(
"fld1\n"
"fpatan\n"
: "=t"(ret)
: "0"(value));
return ret;
#else
# if defined(AK_OS_MINERVA)
return asin(value / sqrt(1 + value * value));
# endif
return __builtin_atan(value);
#endif
}
template<FloatingPoint T>
constexpr T atan2(T y, T x)
{
CONSTEXPR_STATE(atan2, y, x);
#if ARCH(X86_64)
T ret;
asm("fpatan"
: "=t"(ret)
: "0"(x), "u"(y)
: "st(1)");
return ret;
#else
# if defined(AK_OS_MINERVA)
if (__builtin_isnan(y))
return y;
if (__builtin_isnan(x))
return x;
// SPECIAL VALUES
// atan2(±0, -0) returns ±pi.
if (y == 0 && x == 0 && signbit(x))
return copysign(Pi<T>, y);
// atan2(±0, +0) returns ±0.
if (y == 0 && x == 0 && !signbit(x))
return y;
// atan2(±0, x) returns ±pi for x < 0.
if (y == 0 && x < 0)
return copysign(Pi<T>, y);
// atan2(±0, x) returns ±0 for x > 0.
if (y == 0 && x > 0)
return y;
// atan2(y, ±0) returns +pi/2 for y > 0.
if (y > 0 && x == 0)
return Pi<T> / 2;
// atan2(y, ±0) returns -pi/2 for y < 0.
if (y < 0 && x == 0)
return -Pi<T> / 2;
// atan2(±y, -infinity) returns ±pi for finite y > 0.
if (!__builtin_isinf(y) && y > 0 && __builtin_isinf(x) && signbit(x))
return copysign(Pi<T>, y);
// atan2(±y, +infinity) returns ±0 for finite y > 0.
if (!__builtin_isinf(y) && y > 0 && __builtin_isinf(x) && !signbit(x))
return copysign(static_cast<T>(0), y);
// atan2(±infinity, x) returns ±pi/2 for finite x.
if (__builtin_isinf(y) && !__builtin_isinf(x))
return copysign(Pi<T> / 2, y);
// atan2(±infinity, -infinity) returns ±3*pi/4.
if (__builtin_isinf(y) && __builtin_isinf(x) && signbit(x))
return copysign(3 * Pi<T> / 4, y);
// atan2(±infinity, +infinity) returns ±pi/4.
if (__builtin_isinf(y) && __builtin_isinf(x) && !signbit(x))
return copysign(Pi<T> / 4, y);
// Check quadrant, going counterclockwise.
if (y > 0 && x > 0)
return atan(y / x);
if (y > 0 && x < 0)
return atan(y / x) + Pi<T>;
if (y < 0 && x < 0)
return atan(y / x) - Pi<T>;
// y < 0 && x > 0
return atan(y / x);
# else
return __builtin_atan2(y, x);
# endif
#endif
}
}
using Trigonometry::acos;
using Trigonometry::asin;
using Trigonometry::atan;
using Trigonometry::atan2;
using Trigonometry::cos;
using Trigonometry::hypot;
using Trigonometry::sin;
using Trigonometry::sincos;
using Trigonometry::tan;
namespace Exponentials {
template<FloatingPoint T>
constexpr T log2(T x)
{
CONSTEXPR_STATE(log2, x);
#if ARCH(X86_64)
if constexpr (IsSame<T, long double>) {
T ret;
asm(
"fld1\n"
"fxch %%st(1)\n"
"fyl2x\n"
: "=t"(ret)
: "0"(x));
return ret;
}
#endif
// References:
// Gist comparing some implementations
// * https://gist.github.com/Hendiadyoin1/f58346d66637deb9156ef360aa158bf9
if (x == 0)
return -Infinity<T>;
if (x <= 0 || __builtin_isnan(x))
return NaN<T>;
auto ext = FloatExtractor<T>::from_float(x);
T exponent = ext.exponent - FloatExtractor<T>::exponent_bias;
// When the mantissa shows 0b00 (implicitly 1.0) we are on a power of 2
if (ext.mantissa == 0)
return exponent;
// FIXME: Handle denormalized numbers separately
FloatExtractor<T> mantissa_ext {
.mantissa = ext.mantissa,
.exponent = FloatExtractor<T>::exponent_bias,
.sign = ext.sign
};
// (1 <= mantissa < 2)
T m = mantissa_ext.to_float();
// This is a reconstruction of one of Sun's algorithms
// They use a transformation to lower the problem space,
// while keeping the precision, and a 14th degree polynomial,
// which is mirrored at sqrt(2)
// TODO: Sun has some more algorithms for this and other math functions,
// leveraging LUTs, investigate those, if they are better in performance and/or precision.
// These seem to be related to crLibM's implementations, for which papers and references
// are available.
// FIXME: Dynamically adjust the amount of precision between floats and doubles
// AKA floats might need less accuracy here, in comparison to doubles
bool inverted = false;
// m > sqrt(2)
if (m > Sqrt2<T>) {
inverted = true;
m = 2 / m;
}
T s = (m - 1) / (m + 1);
// ln((1 + s) / (1 - s)) == ln(m)
T s2 = s * s;
// clang-format off
T high_approx = s2 * (static_cast<T>(0.6666666666666735130)
+ s2 * (static_cast<T>(0.3999999999940941908)
+ s2 * (static_cast<T>(0.2857142874366239149)
+ s2 * (static_cast<T>(0.2222219843214978396)
+ s2 * (static_cast<T>(0.1818357216161805012)
+ s2 * (static_cast<T>(0.1531383769920937332)
+ s2 * static_cast<T>(0.1479819860511658591)))))));
// clang-format on
// ln(m) == 2 * s + s * high_approx
// log2(m) == log2(e) * ln(m)
T log2_mantissa = L2_E<T> * (2 * s + s * high_approx);
if (inverted)
log2_mantissa = 1 - log2_mantissa;
return exponent + log2_mantissa;
}
template<FloatingPoint T>
constexpr T log(T x)
{
CONSTEXPR_STATE(log, x);
#if ARCH(X86_64)
T ret;
asm(
"fldln2\n"
"fxch %%st(1)\n"
"fyl2x\n"
: "=t"(ret)
: "0"(x));
return ret;
#elif defined(AK_OS_MINERVA)
// FIXME: Adjust the polynomial and formula in log2 to fit this
return log2<T>(x) / L2_E<T>;
#else
return __builtin_log(x);
#endif
}
template<FloatingPoint T>
constexpr T log10(T x)
{
CONSTEXPR_STATE(log10, x);
#if ARCH(X86_64)
T ret;
asm(
"fldlg2\n"
"fxch %%st(1)\n"
"fyl2x\n"
: "=t"(ret)
: "0"(x));
return ret;
#elif defined(AK_OS_MINERVA)
// FIXME: Adjust the polynomial and formula in log2 to fit this
return log2<T>(x) / L2_10<T>;
#else
return __builtin_log10(x);
#endif
}
template<FloatingPoint T>
constexpr T exp2(T exponent)
{
CONSTEXPR_STATE(exp2, exponent);
#if ARCH(X86_64)
T res;
asm("fld1\n"
"fld %%st(1)\n"
"fprem\n"
"f2xm1\n"
"faddp\n"
"fscale\n"
"fstp %%st(1)"
: "=t"(res)
: "0"(exponent));
return res;
#else
// TODO: Add better implementation of this function.
// This is just fast exponentiation for the integer part and
// the first couple terms of the taylor series for the fractional part.
if (exponent < 0)
return 1 / exp2(-exponent);
if (exponent >= log2(NumericLimits<T>::max()))
return Infinity<T>;
// Integer exponentiation part.
int int_exponent = static_cast<int>(exponent);
T exponent_fraction = exponent - int_exponent;
T int_result = 1;
T base = 2;
for (;;) {
if (int_exponent & 1)
int_result *= base;
int_exponent >>= 1;
if (!int_exponent)
break;
base *= base;
}
// Fractional part.
// Uses:
// exp(x) = sum(n, 0, \infty, x ** n / n!)
// 2**x = exp(log2(e) * x)
// FIXME: Pick better step size (and make it dependent on T).
T result = 0;
T power = 1;
T factorial = 1;
for (int i = 1; i < 16; ++i) {
result += power / factorial;
power *= exponent_fraction / L2_E<T>;
factorial *= i;
}
return int_result * result;
#endif
}
template<FloatingPoint T>
constexpr T exp(T exponent)
{
CONSTEXPR_STATE(exp, exponent);
#if ARCH(X86_64)
T res;
asm("fldl2e\n"
"fmulp\n"
"fld1\n"
"fld %%st(1)\n"
"fprem\n"
"f2xm1\n"
"faddp\n"
"fscale\n"
"fstp %%st(1)"
: "=t"(res)
: "0"(exponent));
return res;
#else
// TODO: Add better implementation of this function.
return exp2(exponent * L2_E<T>);
#endif
}
}
using Exponentials::exp;
using Exponentials::exp2;
using Exponentials::log;
using Exponentials::log10;
using Exponentials::log2;
namespace Hyperbolic {
template<FloatingPoint T>
constexpr T sinh(T x)
{
T exponentiated = exp<T>(x);
if (x > 0)
return (exponentiated * exponentiated - 1) / 2 / exponentiated;
return (exponentiated - 1 / exponentiated) / 2;
}
template<FloatingPoint T>
constexpr T cosh(T x)
{
CONSTEXPR_STATE(cosh, x);
T exponentiated = exp(-x);
if (x < 0)
return (1 + exponentiated * exponentiated) / 2 / exponentiated;
return (1 / exponentiated + exponentiated) / 2;
}
template<FloatingPoint T>
constexpr T tanh(T x)
{
if (x > 0) {
T exponentiated = exp<T>(2 * x);
return (exponentiated - 1) / (exponentiated + 1);
}
T plusX = exp<T>(x);
T minusX = 1 / plusX;
return (plusX - minusX) / (plusX + minusX);
}
template<FloatingPoint T>
constexpr T asinh(T x)
{
return log<T>(x + sqrt<T>(x * x + 1));
}
template<FloatingPoint T>
constexpr T acosh(T x)
{
return log<T>(x + sqrt<T>(x * x - 1));
}
template<FloatingPoint T>
constexpr T atanh(T x)
{
return log<T>((1 + x) / (1 - x)) / (T)2.0l;
}
}
using Hyperbolic::acosh;
using Hyperbolic::asinh;
using Hyperbolic::atanh;
using Hyperbolic::cosh;
using Hyperbolic::sinh;
using Hyperbolic::tanh;
// Calculate x^y with fast exponentiation when the power is a natural number.
template<FloatingPoint F, Unsigned U>
constexpr F pow_int(F x, U y)
{
auto result = static_cast<F>(1);
while (y > 0) {
if (y % 2 == 1) {
result *= x;
}
x = x * x;
y >>= 1;
}
return result;
}
template<FloatingPoint T>
constexpr T pow(T x, T y)
{
CONSTEXPR_STATE(pow, x, y);
if (__builtin_isnan(y))
return y;
if (y == 0)
return 1;
if (x == 0)
return 0;
if (y == 1)
return x;
// Take an integer fast path as long as the value fits within a 64-bit integer.
if (y >= static_cast<T>(NumericLimits<i64>::min()) && y < static_cast<T>(NumericLimits<i64>::max())) {
i64 y_as_int = static_cast<i64>(y);
if (y == static_cast<T>(y_as_int)) {
T result = pow_int(x, static_cast<u64>(fabs<T>(y)));
if (y < 0)
result = static_cast<T>(1.0l) / result;
return result;
}
}
// FIXME: This formula suffers from error magnification.
return exp2<T>(y * log2<T>(x));
}
template<Integral I, typename T>
constexpr I clamp_to(T value)
{
constexpr auto max = static_cast<T>(NumericLimits<I>::max());
if constexpr (max > 0) {
if (value >= static_cast<T>(NumericLimits<I>::max()))
return NumericLimits<I>::max();
}
constexpr auto min = static_cast<T>(NumericLimits<I>::min());
if constexpr (min <= 0) {
if (value <= static_cast<T>(NumericLimits<I>::min()))
return NumericLimits<I>::min();
}
if constexpr (IsFloatingPoint<T>)
return round_to<I>(value);
return value;
}
#undef CONSTEXPR_STATE
#undef AARCH64_INSTRUCTION
}
#if USING_AK_GLOBALLY
using AK::round_to;
#endif
