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flst / upstream /emscripten /system /lib /llvm-libc /src /math /generic /acos.cpp
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//===-- Double-precision acos function ------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#include "src/math/acos.h"
#include "asin_utils.h"
#include "src/__support/FPUtil/FEnvImpl.h"
#include "src/__support/FPUtil/FPBits.h"
#include "src/__support/FPUtil/PolyEval.h"
#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/FPUtil/sqrt.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
namespace LIBC_NAMESPACE_DECL {
using DoubleDouble = fputil::DoubleDouble;
using Float128 = fputil::DyadicFloat<128>;
LLVM_LIBC_FUNCTION(double, acos, (double x)) {
using FPBits = fputil::FPBits<double>;
FPBits xbits(x);
int x_exp = xbits.get_biased_exponent();
// |x| < 0.5.
if (x_exp < FPBits::EXP_BIAS - 1) {
// |x| < 2^-55.
if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 55)) {
// When |x| < 2^-55, acos(x) = pi/2
#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)
return PI_OVER_TWO.hi;
#else
// Force the evaluation and prevent constant propagation so that it
// is rounded correctly for FE_UPWARD rounding mode.
return (xbits.abs().get_val() + 0x1.0p-160) + PI_OVER_TWO.hi;
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
// acos(x) = pi/2 - asin(x)
// = pi/2 - x * P(x^2)
double p = asin_eval(x * x);
return PI_OVER_TWO.hi + fputil::multiply_add(-x, p, PI_OVER_TWO.lo);
#else
unsigned idx;
DoubleDouble x_sq = fputil::exact_mult(x, x);
double err = xbits.abs().get_val() * 0x1.0p-51;
// Polynomial approximation:
// p ~ asin(x)/x
DoubleDouble p = asin_eval(x_sq, idx, err);
// asin(x) ~ x * p
DoubleDouble r0 = fputil::exact_mult(x, p.hi);
// acos(x) = pi/2 - asin(x)
// ~ pi/2 - x * p
// = pi/2 - x * (p.hi + p.lo)
double r_hi = fputil::multiply_add(-x, p.hi, PI_OVER_TWO.hi);
// Use Dekker's 2SUM algorithm to compute the lower part.
double r_lo = ((PI_OVER_TWO.hi - r_hi) - r0.hi) - r0.lo;
r_lo = fputil::multiply_add(-x, p.lo, r_lo + PI_OVER_TWO.lo);
// Ziv's accuracy test.
double r_upper = r_hi + (r_lo + err);
double r_lower = r_hi + (r_lo - err);
if (LIBC_LIKELY(r_upper == r_lower))
return r_upper;
// Ziv's accuracy test failed, perform 128-bit calculation.
// Recalculate mod 1/64.
idx = static_cast<unsigned>(fputil::nearest_integer(x_sq.hi * 0x1.0p6));
// Get x^2 - idx/64 exactly. When FMA is available, double-double
// multiplication will be correct for all rounding modes. Otherwise we use
// Float128 directly.
Float128 x_f128(x);
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// u = x^2 - idx/64
Float128 u_hi(
fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi));
Float128 u = fputil::quick_add(u_hi, Float128(x_sq.lo));
#else
Float128 x_sq_f128 = fputil::quick_mul(x_f128, x_f128);
Float128 u = fputil::quick_add(
x_sq_f128, Float128(static_cast<double>(idx) * (-0x1.0p-6)));
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
Float128 p_f128 = asin_eval(u, idx);
// Flip the sign of x_f128 to perform subtraction.
x_f128.sign = x_f128.sign.negate();
Float128 r =
fputil::quick_add(PI_OVER_TWO_F128, fputil::quick_mul(x_f128, p_f128));
return static_cast<double>(r);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
// |x| >= 0.5
double x_abs = xbits.abs().get_val();
// Maintaining the sign:
constexpr double SIGN[2] = {1.0, -1.0};
double x_sign = SIGN[xbits.is_neg()];
// |x| >= 1
if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) {
// x = +-1, asin(x) = +- pi/2
if (x_abs == 1.0) {
// x = 1, acos(x) = 0,
// x = -1, acos(x) = pi
return x == 1.0 ? 0.0 : fputil::multiply_add(-x_sign, PI.hi, PI.lo);
}
// |x| > 1, return NaN.
if (xbits.is_quiet_nan())
return x;
// Set domain error for non-NaN input.
if (!xbits.is_nan())
fputil::set_errno_if_required(EDOM);
fputil::raise_except_if_required(FE_INVALID);
return FPBits::quiet_nan().get_val();
}
// When |x| >= 0.5, we perform range reduction as follow:
//
// When 0.5 <= x < 1, let:
// y = acos(x)
// We will use the double angle formula:
// cos(2y) = 1 - 2 sin^2(y)
// and the complement angle identity:
// x = cos(y) = 1 - 2 sin^2 (y/2)
// So:
// sin(y/2) = sqrt( (1 - x)/2 )
// And hence:
// y/2 = asin( sqrt( (1 - x)/2 ) )
// Equivalently:
// acos(x) = y = 2 * asin( sqrt( (1 - x)/2 ) )
// Let u = (1 - x)/2, then:
// acos(x) = 2 * asin( sqrt(u) )
// Moreover, since 0.5 <= x < 1:
// 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5,
// And hence we can reuse the same polynomial approximation of asin(x) when
// |x| <= 0.5:
// acos(x) ~ 2 * sqrt(u) * P(u).
//
// When -1 < x <= -0.5, we reduce to the previous case using the formula:
// acos(x) = pi - acos(-x)
// = pi - 2 * asin ( sqrt( (1 + x)/2 ) )
// ~ pi - 2 * sqrt(u) * P(u),
// where u = (1 - |x|)/2.
// u = (1 - |x|)/2
double u = fputil::multiply_add(x_abs, -0.5, 0.5);
// v_hi + v_lo ~ sqrt(u).
// Let:
// h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi)
// Then:
// sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
// ~ v_hi + h / (2 * v_hi)
// So we can use:
// v_lo = h / (2 * v_hi).
double v_hi = fputil::sqrt<double>(u);
#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
constexpr DoubleDouble CONST_TERM[2] = {{0.0, 0.0}, PI};
DoubleDouble const_term = CONST_TERM[xbits.is_neg()];
double p = asin_eval(u);
double scale = x_sign * 2.0 * v_hi;
double r = const_term.hi + fputil::multiply_add(scale, p, const_term.lo);
return r;
#else
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
double h = fputil::multiply_add(v_hi, -v_hi, u);
#else
DoubleDouble v_hi_sq = fputil::exact_mult(v_hi, v_hi);
double h = (u - v_hi_sq.hi) - v_hi_sq.lo;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// Scale v_lo and v_hi by 2 from the formula:
// vh = v_hi * 2
// vl = 2*v_lo = h / v_hi.
double vh = v_hi * 2.0;
double vl = h / v_hi;
// Polynomial approximation:
// p ~ asin(sqrt(u))/sqrt(u)
unsigned idx;
double err = vh * 0x1.0p-51;
DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err);
// Perform computations in double-double arithmetic:
// asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)
DoubleDouble r0 = fputil::quick_mult(DoubleDouble{vl, vh}, p);
double r_hi, r_lo;
if (xbits.is_pos()) {
r_hi = r0.hi;
r_lo = r0.lo;
} else {
DoubleDouble r = fputil::exact_add(PI.hi, -r0.hi);
r_hi = r.hi;
r_lo = (PI.lo - r0.lo) + r.lo;
}
// Ziv's accuracy test.
double r_upper = r_hi + (r_lo + err);
double r_lower = r_hi + (r_lo - err);
if (LIBC_LIKELY(r_upper == r_lower))
return r_upper;
// Ziv's accuracy test failed, we redo the computations in Float128.
// Recalculate mod 1/64.
idx = static_cast<unsigned>(fputil::nearest_integer(u * 0x1.0p6));
// After the first step of Newton-Raphson approximating v = sqrt(u), we have
// that:
// sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
// v_lo = h / (2 * v_hi)
// With error:
// sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )
// = -h^2 / (2*v * (sqrt(u) + v)^2).
// Since:
// (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,
// we can add another correction term to (v_hi + v_lo) that is:
// v_ll = -h^2 / (2*v_hi * 4u)
// = -v_lo * (h / 4u)
// = -vl * (h / 8u),
// making the errors:
// sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)
// well beyond 128-bit precision needed.
// Get the rounding error of vl = 2 * v_lo ~ h / vh
// Get full product of vh * vl
#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;
#else
DoubleDouble vh_vl = fputil::exact_mult(v_hi, vl);
double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;
#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
// vll = 2*v_ll = -vl * (h / (4u)).
double t = h * (-0.25) / u;
double vll = fputil::multiply_add(vl, t, vl_lo);
// m_v = -(v_hi + v_lo + v_ll).
Float128 m_v = fputil::quick_add(
Float128(vh), fputil::quick_add(Float128(vl), Float128(vll)));
m_v.sign = xbits.sign();
// Perform computations in Float128:
// acos(x) = (v_hi + v_lo + vll) * P(u) , when 0.5 <= x < 1,
// = pi - (v_hi + v_lo + vll) * P(u) , when -1 < x <= -0.5.
Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u));
Float128 p_f128 = asin_eval(y_f128, idx);
Float128 r_f128 = fputil::quick_mul(m_v, p_f128);
if (xbits.is_neg())
r_f128 = fputil::quick_add(PI_F128, r_f128);
return static_cast<double>(r_f128);
#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
}
} // namespace LIBC_NAMESPACE_DECL