mirror of
https://github.com/SerenityOS/serenity
synced 2026-04-25 17:15:42 +02:00
65209b7a9a
sqrt, sin, cos, tan, atan, atan2, log, log10 used to always call the double built-in. Now the float overload calls the float built-in, the double overload the double built-in, and the long double overload the long double built-in. Ideally, we'd stop calling built-ins for these (see #26662), but as long as we do, we should call the right ones. Similar to the last two commits in #18998.
1383 lines
36 KiB
C++
1383 lines
36 KiB
C++
/*
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* Copyright (c) 2021, Leon Albrecht <leon2002.la@gmail.com>.
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*
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* SPDX-License-Identifier: BSD-2-Clause
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*/
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#pragma once
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#include <AK/BuiltinWrappers.h>
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#include <AK/Concepts.h>
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#include <AK/FloatingPoint.h>
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#include <AK/NumericLimits.h>
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#include <AK/StdLibExtraDetails.h>
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#include <AK/Types.h>
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#include <math.h>
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#ifdef KERNEL
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# error "Including AK/Math.h from the Kernel is never correct! Floating point is disabled."
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#endif
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namespace AK {
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template<FloatingPoint T>
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constexpr T NaN = __builtin_nan("");
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template<FloatingPoint T>
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constexpr T Infinity = __builtin_huge_vall();
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template<FloatingPoint T>
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constexpr T Pi = 3.141592653589793238462643383279502884L;
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template<FloatingPoint T>
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constexpr T E = 2.718281828459045235360287471352662498L;
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template<FloatingPoint T>
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constexpr T Sqrt2 = 1.414213562373095048801688724209698079L;
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template<FloatingPoint T>
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constexpr T Sqrt1_2 = 0.707106781186547524400844362104849039L;
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template<FloatingPoint T>
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constexpr T L2_10 = 3.321928094887362347870319429489390175864L;
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template<FloatingPoint T>
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constexpr T L2_E = 1.442695040888963407359924681001892137L;
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namespace Details {
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template<size_t>
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constexpr size_t product_even();
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template<>
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constexpr size_t product_even<2>() { return 2; }
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template<size_t value>
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constexpr size_t product_even() { return value * product_even<value - 2>(); }
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template<size_t>
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constexpr size_t product_odd();
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template<>
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constexpr size_t product_odd<1>() { return 1; }
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template<size_t value>
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constexpr size_t product_odd() { return value * product_odd<value - 2>(); }
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}
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template<FloatingPoint T>
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constexpr T to_radians(T degrees)
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{
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return degrees * AK::Pi<T> / 180;
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}
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template<FloatingPoint T>
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constexpr T to_degrees(T radians)
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{
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return radians * 180 / AK::Pi<T>;
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}
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template<FloatingPoint FloatT>
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FloatT copysign(FloatT x, FloatT y)
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{
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using Extractor = FloatExtractor<FloatT>;
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auto ex = Extractor::from_float(x);
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auto ey = Extractor::from_float(y);
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ex.sign = ey.sign;
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return ex.to_float();
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}
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#define CALL_BUILTIN(function, args...) \
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do { \
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if constexpr (IsSame<T, long double>) \
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return __builtin_##function##l(args); \
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if constexpr (IsSame<T, double>) \
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return __builtin_##function(args); \
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if constexpr (IsSame<T, float>) \
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return __builtin_##function##f(args); \
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} while (0)
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#define CONSTEXPR_STATE(function, args...) \
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if (is_constant_evaluated()) \
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CALL_BUILTIN(function, args);
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#define AARCH64_INSTRUCTION(instruction, arg) \
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if constexpr (IsSame<T, long double>) \
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TODO(); \
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if constexpr (IsSame<T, double>) { \
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double res; \
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asm(#instruction " %d0, %d1" \
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: "=w"(res) \
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: "w"(arg)); \
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return res; \
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} \
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if constexpr (IsSame<T, float>) { \
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float res; \
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asm(#instruction " %s0, %s1" \
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: "=w"(res) \
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: "w"(arg)); \
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return res; \
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}
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template<FloatingPoint T>
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constexpr T fabs(T x)
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{
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// Both GCC and Clang inline fabs by default, so this is just a cmath like wrapper
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CALL_BUILTIN(fabs, x);
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}
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namespace Rounding {
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template<FloatingPoint T>
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constexpr T ceil(T num)
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{
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// FIXME: SSE4.1 rounds[sd] num, res, 0b110
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if (is_constant_evaluated()) {
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if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
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return num;
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return (static_cast<T>(static_cast<i64>(num)) == num)
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? static_cast<i64>(num)
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: static_cast<i64>(num) + ((num > 0) ? 1 : 0);
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}
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#if ARCH(AARCH64)
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AARCH64_INSTRUCTION(frintp, num);
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#else
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CALL_BUILTIN(ceil, num);
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#endif
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}
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template<FloatingPoint T>
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constexpr T floor(T num)
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{
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// FIXME: SSE4.1 rounds[sd] num, res, 0b101
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if (is_constant_evaluated()) {
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if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
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return num;
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return (static_cast<T>(static_cast<i64>(num)) == num)
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? static_cast<i64>(num)
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: static_cast<i64>(num) - ((num > 0) ? 0 : 1);
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}
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#if ARCH(AARCH64)
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AARCH64_INSTRUCTION(frintm, num);
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#else
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CALL_BUILTIN(floor, num);
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#endif
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}
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template<FloatingPoint T>
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constexpr T trunc(T num)
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{
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#if ARCH(AARCH64)
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if (is_constant_evaluated()) {
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if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
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return num;
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return static_cast<T>(static_cast<i64>(num));
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}
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AARCH64_INSTRUCTION(frintz, num);
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#endif
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// FIXME: Use dedicated instruction in the non constexpr case
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// SSE4.1: rounds[sd] %num, %res, 0b111
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if (num < NumericLimits<i64>::min() || num > NumericLimits<i64>::max())
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return num;
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return static_cast<T>(static_cast<i64>(num));
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}
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template<FloatingPoint T>
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constexpr T rint(T x)
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{
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CONSTEXPR_STATE(rint, x);
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// Note: This does break tie to even
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// But the behavior of frndint/rounds[ds]/frintx can be configured
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// through the floating point control registers.
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// FIXME: We should decide if we rename this to allow us to get away from
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// the configurability "burden" rint has
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// this would make us use `rounds[sd] %num, %res, 0b100`
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// and `frintn` respectively,
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// no such guaranteed round exists for x87 `frndint`
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#if ARCH(X86_64)
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# ifdef __SSE4_1__
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if constexpr (IsSame<T, double>) {
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T r;
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asm(
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"roundsd %1, %0"
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: "=x"(r)
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: "x"(x));
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return r;
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}
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if constexpr (IsSame<T, float>) {
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T r;
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asm(
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"roundss %1, %0"
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: "=x"(r)
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: "x"(x));
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return r;
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}
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# else
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asm(
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"frndint"
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: "+t"(x));
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return x;
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# endif
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#elif ARCH(AARCH64)
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AARCH64_INSTRUCTION(frintx, x);
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#elif ARCH(RISCV64)
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if (__builtin_isnan(x))
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return x;
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// Floating point values have a gap size of >= 1 for values above 2^mantissa_bits - 1.
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if (fabs(x) > FloatExtractor<T>::mantissa_max)
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return x;
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if constexpr (IsSame<T, float>) {
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i64 r;
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asm("fcvt.l.s %0, %1, dyn"
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: "=r"(r)
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: "f"(x));
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return copysign(static_cast<float>(r), x);
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}
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if constexpr (IsSame<T, double>) {
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i64 r;
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asm("fcvt.l.d %0, %1, dyn"
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: "=r"(r)
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: "f"(x));
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return copysign(static_cast<double>(r), x);
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}
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if constexpr (IsSame<T, long double>)
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TODO_RISCV64();
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#endif
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TODO();
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}
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template<FloatingPoint T>
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constexpr T round(T x)
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{
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CONSTEXPR_STATE(round, x);
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// Note: This is break-tie-away-from-zero, so not the hw's understanding of
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// "nearest", which would be towards even.
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if (x == 0.)
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return x;
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if (x > 0.)
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return floor(x + .5);
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return ceil(x - .5);
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}
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template<Integral I, FloatingPoint P>
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ALWAYS_INLINE I round_to(P value);
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#if ARCH(X86_64)
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template<Integral I>
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ALWAYS_INLINE I round_to(long double value)
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{
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// Note: fistps outputs into a signed integer location (i16, i32, i64),
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// so lets be nice and tell the compiler that.
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Conditional<sizeof(I) >= sizeof(i16), MakeSigned<I>, i16> ret;
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if constexpr (sizeof(I) == sizeof(i64)) {
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asm("fistpll %0"
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: "=m"(ret)
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: "t"(value)
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: "st");
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} else if constexpr (sizeof(I) == sizeof(i32)) {
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asm("fistpl %0"
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: "=m"(ret)
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: "t"(value)
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: "st");
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} else {
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asm("fistps %0"
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: "=m"(ret)
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: "t"(value)
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: "st");
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}
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return static_cast<I>(ret);
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}
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template<Integral I>
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ALWAYS_INLINE I round_to(float value)
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{
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// FIXME: round_to<u64> might will cause issues, aka the indefinite value being set,
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// if the value surpasses the i64 limit, even if the result could fit into an u64
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// To solve this we would either need to detect that value or do a range check and
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// then do a more specialized conversion, which might include a division (which is expensive)
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if constexpr (sizeof(I) == sizeof(i64) || IsSame<I, u32>) {
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i64 ret;
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asm("cvtss2si %1, %0"
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: "=r"(ret)
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: "xm"(value));
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return static_cast<I>(ret);
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}
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i32 ret;
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asm("cvtss2si %1, %0"
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: "=r"(ret)
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: "xm"(value));
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return static_cast<I>(ret);
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}
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template<Integral I>
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ALWAYS_INLINE I round_to(double value)
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{
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// FIXME: round_to<u64> might will cause issues, aka the indefinite value being set,
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// if the value surpasses the i64 limit, even if the result could fit into an u64
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// To solve this we would either need to detect that value or do a range check and
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// then do a more specialized conversion, which might include a division (which is expensive)
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if constexpr (sizeof(I) == sizeof(i64) || IsSame<I, u32>) {
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i64 ret;
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asm("cvtsd2si %1, %0"
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: "=r"(ret)
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: "xm"(value));
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return static_cast<I>(ret);
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}
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i32 ret;
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asm("cvtsd2si %1, %0"
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: "=r"(ret)
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: "xm"(value));
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return static_cast<I>(ret);
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}
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#elif ARCH(AARCH64)
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template<SignedIntegral I>
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ALWAYS_INLINE I round_to(float value)
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{
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if constexpr (sizeof(I) <= sizeof(u32)) {
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i32 res;
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asm("fcvtns %w0, %s1"
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: "=r"(res)
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: "w"(value));
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return static_cast<I>(res);
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}
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i64 res;
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asm("fcvtns %0, %s1"
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: "=r"(res)
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: "w"(value));
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return static_cast<I>(res);
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}
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template<SignedIntegral I>
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ALWAYS_INLINE I round_to(double value)
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{
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if constexpr (sizeof(I) <= sizeof(u32)) {
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i32 res;
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asm("fcvtns %w0, %d1"
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: "=r"(res)
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: "w"(value));
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return static_cast<I>(res);
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}
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i64 res;
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asm("fcvtns %0, %d1"
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: "=r"(res)
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: "w"(value));
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return static_cast<I>(res);
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}
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template<UnsignedIntegral U>
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ALWAYS_INLINE U round_to(float value)
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{
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if constexpr (sizeof(U) <= sizeof(u32)) {
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u32 res;
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asm("fcvtnu %w0, %s1"
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: "=r"(res)
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: "w"(value));
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return static_cast<U>(res);
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}
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i64 res;
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asm("fcvtnu %0, %s1"
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: "=r"(res)
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: "w"(value));
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return static_cast<U>(res);
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}
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template<UnsignedIntegral U>
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ALWAYS_INLINE U round_to(double value)
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{
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if constexpr (sizeof(U) <= sizeof(u32)) {
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u32 res;
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asm("fcvtns %w0, %d1"
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: "=r"(res)
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: "w"(value));
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return static_cast<U>(res);
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}
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i64 res;
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asm("fcvtns %0, %d1"
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: "=r"(res)
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: "w"(value));
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return static_cast<U>(res);
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}
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#else
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template<Integral I, FloatingPoint P>
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ALWAYS_INLINE I round_to(P value)
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{
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if constexpr (IsSame<P, long double>)
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return static_cast<I>(__builtin_llrintl(value));
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if constexpr (IsSame<P, double>)
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return static_cast<I>(__builtin_llrint(value));
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if constexpr (IsSame<P, float>)
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return static_cast<I>(__builtin_llrintf(value));
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}
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#endif
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}
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using Rounding::ceil;
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using Rounding::floor;
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using Rounding::rint;
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using Rounding::round;
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using Rounding::round_to;
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using Rounding::trunc;
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namespace Division {
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template<FloatingPoint T>
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constexpr T fmod(T x, T y)
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{
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CONSTEXPR_STATE(fmod, x, y);
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#if ARCH(X86_64)
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u16 fpu_status;
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do {
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asm(
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"fprem\n"
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"fnstsw %%ax\n"
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: "+t"(x), "=a"(fpu_status)
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: "u"(y));
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} while (fpu_status & 0x400);
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return x;
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#else
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# if defined(AK_OS_SERENITY)
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// FIXME: This is a very naive implementation.
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if (__builtin_isnan(x))
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return x;
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if (__builtin_isnan(y))
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return y;
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// SPECIAL VALUES
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// fmod(±0, y) returns ±0 if y is neither 0 nor NaN.
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if (x == 0 && y != 0)
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return x;
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// fmod(x, y) returns a NaN and raises the "invalid" floating-point exception for x infinite or y zero.
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// FIXME: Exception.
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if (__builtin_isinf(x) || y == 0)
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return NaN<T>;
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// fmod(x, ±infinity) returns x for x not infinite.
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if (__builtin_isinf(y))
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return x;
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// Range reduction: handle negative x and y.
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if (y < 0)
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return fmod(x, -y);
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if (x < 0)
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return -fmod(-x, y);
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// x is x_mantissa * 2**x_exponent, y is y_mantissa * 2**y_exponent.
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// (Both are positive at this point.)
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// If y_exponent > x_exponent, we are done and can return x.
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// If y_exponent == x_exponent, we can return (x_mantissa % y_mantissa) * 2**x_exponent.
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// If y_exponent < x_exponent, we'll iteratively reduce x_exponent by shifting from
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// the exponent into the mantissa.
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auto x_bits = FloatExtractor<T>::from_float(x);
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typename FloatExtractor<T>::ComponentType x_exponent = x_bits.exponent;
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auto y_bits = FloatExtractor<T>::from_float(y);
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typename FloatExtractor<T>::ComponentType y_exponent = y_bits.exponent;
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// FIXME: Handle denormals. For now, treat them as 0.
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if (x_exponent == 0 && y_exponent != 0)
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return 0;
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if (y_exponent == 0)
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return NaN<T>;
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if (y_exponent > x_exponent)
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return x;
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// FIXME: This is wrong for f80.
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typename FloatExtractor<T>::ComponentType implicit_mantissa_bit = 1;
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implicit_mantissa_bit <<= FloatExtractor<T>::mantissa_bits;
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typename FloatExtractor<T>::ComponentType x_mantissa = x_bits.mantissa | implicit_mantissa_bit;
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typename FloatExtractor<T>::ComponentType y_mantissa = y_bits.mantissa | implicit_mantissa_bit;
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while (y_exponent < x_exponent) {
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// This is ok because (x % (y * 2**n)) divides (x % y) for all n > 0.
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x_mantissa %= y_mantissa;
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x_mantissa <<= 1;
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--x_exponent;
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}
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x_mantissa %= y_mantissa;
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if (x_mantissa == 0)
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|
return 0;
|
|
|
|
// 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
|
|
CALL_BUILTIN(fmod, 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_SERENITY)
|
|
// TODO: Add implementation for this function.
|
|
TODO();
|
|
# endif
|
|
CALL_BUILTIN(remainder, 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_SERENITY)
|
|
// TODO: Add implementation for this function.
|
|
TODO();
|
|
# endif
|
|
CALL_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_SERENITY)
|
|
T sign = 1;
|
|
if (angle < 0) {
|
|
angle = -angle;
|
|
sign = -1;
|
|
}
|
|
|
|
if (angle >= 2 * Pi<T>)
|
|
angle = fmod(angle, 2 * Pi<T>);
|
|
|
|
if (angle >= Pi<T>) {
|
|
angle = angle - Pi<T>;
|
|
sign = -sign;
|
|
}
|
|
|
|
if (angle > Pi<T> / 2)
|
|
angle = Pi<T> - angle;
|
|
|
|
// https://github.com/samhocevar/lolremez/wiki/Tutorial-4-of-5%3A-fixing-lower-order-parameters
|
|
auto f = [](T x) {
|
|
if constexpr (IsSame<T, f32>) {
|
|
// lolremez --float --degree 3 --range "1e-50:pi*pi/4" "(sin(sqrt(x))-sqrt(x))/(x*sqrt(x))" "1/(x*sqrt(x))"
|
|
// "Estimated max error: 4.618689e-9"
|
|
f32 u = 2.6000548e-06f;
|
|
u = u * x + -0.00019806615f;
|
|
u = u * x + 0.0083330171f;
|
|
return u * x + -0.16666657f;
|
|
} else {
|
|
// FIXME: Could do something custom for long double.
|
|
// lolremez --degree 6 --range "1e-50:pi*pi/4" "(sin(sqrt(x))-sqrt(x))/(x*sqrt(x))" "1/(x*sqrt(x))"
|
|
// "Estimated max error: 1.1015766629825144e-16"
|
|
T u = -7.3646464502210486e-13;
|
|
u = u * x + 1.6047301196685753e-10;
|
|
u = u * x + -2.5051851497012596e-08;
|
|
u = u * x + 2.7557316077007725e-06;
|
|
u = u * x + -0.00019841269820094207;
|
|
u = u * x + 0.0083333333332628792;
|
|
return u * x + -0.16666666666665811;
|
|
}
|
|
};
|
|
T angle_squared = angle * angle;
|
|
return sign * (angle + angle * angle_squared * f(angle_squared));
|
|
# else
|
|
CALL_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_SERENITY)
|
|
if (angle < 0)
|
|
angle = -angle;
|
|
|
|
if (angle >= 2 * Pi<T>)
|
|
angle = fmod(angle, 2 * Pi<T>);
|
|
|
|
T sign = 1;
|
|
if (angle >= Pi<T>) {
|
|
angle = angle - Pi<T>;
|
|
sign = -1;
|
|
}
|
|
|
|
if (angle > Pi<T> / 2) {
|
|
angle = Pi<T> - angle;
|
|
sign = -sign;
|
|
}
|
|
|
|
// https://github.com/samhocevar/lolremez/wiki/Tutorial-3-of-5:-changing-variables-for-simpler-polynomials
|
|
// for cos(x): cos(x) - 1 is a function of x^2 terms, so we do the substitution on that page like
|
|
// max | cos(x) - 1 - x^2 Q(x^2) | and then set y = x^2. That yields:
|
|
// max | (cos(sqrt(y)) - 1) - y Q(y) |. Dividing through with y (instead of sqrt(y) as in the sin() case) gives us:
|
|
auto f = [](T x) {
|
|
if constexpr (IsSame<T, f32>) {
|
|
// lolremez --float --degree 3 --range "1e-50:pi*pi/4" "(cos(sqrt(x)) - 1)/x" "1/x"
|
|
// "Estimated max error: 5.2720126e-8"
|
|
f32 u = 2.3194387e-05f;
|
|
u = u * x + -0.0013855927f;
|
|
u = u * x + 0.041663989f;
|
|
return u * x + -0.49999931f;
|
|
} else {
|
|
// lolremez --degree 6 --range "1e-50:pi*pi/4" "(cos(sqrt(x)) - 1)/x" "1/x"
|
|
// "Estimated max error: 2.0880269759116624e-15"
|
|
T u = -1.101249182846601e-11;
|
|
u = u * x + 2.0858735176345955e-09;
|
|
u = u * x + -2.7556950755056579e-07;
|
|
u = u * x + 2.4801583156341611e-05;
|
|
u = u * x + -0.001388888886393419;
|
|
u = u * x + 0.041666666665954213;
|
|
return u * x + -0.49999999999993117;
|
|
}
|
|
};
|
|
T angle_squared = angle * angle;
|
|
return sign * (1 + angle_squared * f(angle_squared));
|
|
# else
|
|
CALL_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_SERENITY)
|
|
return sin(angle) / cos(angle);
|
|
# else
|
|
CALL_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_SERENITY)
|
|
if (value < 0)
|
|
return -atan(-value);
|
|
|
|
if (value > 1)
|
|
return Pi<T> / 2 - atan(1 / value);
|
|
|
|
// atan is an odd function a leading factor of 1, so this uses the same substitutions as sin().
|
|
// See there for a description.
|
|
auto f = [](T x) {
|
|
if constexpr (IsSame<T, f32>) {
|
|
// lolremez --float --degree 7 --range "1e-50:1" "(atan(sqrt(x))-sqrt(x))/(x*sqrt(x))" "1/(x*sqrt(x))"
|
|
// "Estimated max error: 7.351572e-9"
|
|
float u = 0.0026222446f;
|
|
u = u * x + -0.015132537f;
|
|
u = u * x + 0.041121863f;
|
|
u = u * x + -0.073667064f;
|
|
u = u * x + 0.10573932f;
|
|
u = u * x + -0.14185975f;
|
|
u = u * x + 0.19990396f;
|
|
return u * x + -0.33332986f;
|
|
} else {
|
|
// FIXME: Could do something custom for long double.
|
|
// lolremez --degree 15 --range "1e-50:1" "(atan(sqrt(x))-sqrt(x))/(x*sqrt(x))" "1/(x*sqrt(x))"
|
|
// "Estimated max error: 2.695747111292741e-15"
|
|
T u = 6.4855700791782353e-05;
|
|
u = u * x + -0.00062980993515420608;
|
|
u = u * x + 0.0028877745234626882;
|
|
u = u * x + -0.0083913659122280861;
|
|
u = u * x + 0.01759373283992496;
|
|
u = u * x + -0.028943865588822337;
|
|
u = u * x + 0.04001711781175539;
|
|
u = u * x + -0.049493567473208426;
|
|
u = u * x + 0.05782092815821073;
|
|
u = u * x + -0.066423996784058609;
|
|
u = u * x + 0.076879768543915233;
|
|
u = u * x + -0.090903598304650779;
|
|
u = u * x + 0.11111064186237864;
|
|
u = u * x + -0.14285711801574916;
|
|
u = u * x + 0.19999999929660117;
|
|
return u * x + -0.33333333332571796;
|
|
}
|
|
};
|
|
T squared = value * value;
|
|
return value + value * squared * f(squared);
|
|
# endif
|
|
CALL_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_SERENITY)
|
|
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
|
|
CALL_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_SERENITY)
|
|
// FIXME: Adjust the polynomial and formula in log2 to fit this
|
|
return log2<T>(x) / L2_E<T>;
|
|
#else
|
|
CALL_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_SERENITY)
|
|
// FIXME: Adjust the polynomial and formula in log2 to fit this
|
|
return log2<T>(x) / L2_10<T>;
|
|
#else
|
|
CALL_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, UnsignedIntegral 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;
|
|
}
|
|
|
|
// Wrap a to keep it in the range [-b, b].
|
|
template<typename T>
|
|
constexpr T wrap_to_range(T a, IdentityType<T> b)
|
|
{
|
|
return fmod(fmod(a + b, 2 * b) + 2 * b, 2 * b) - b;
|
|
}
|
|
|
|
#undef CALL_BUILTIN
|
|
#undef CONSTEXPR_STATE
|
|
#undef AARCH64_INSTRUCTION
|
|
}
|
|
|
|
#if USING_AK_GLOBALLY
|
|
using AK::round_to;
|
|
#endif
|