[libc-commits] [libc] [libc][math] Implement double precision asin correctly rounded for all rounding modes. (PR #134401)
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libc-commits at lists.llvm.org
Mon Apr 14 09:51:20 PDT 2025
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@@ -0,0 +1,284 @@
+//===-- Double-precision asin 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/asin.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, asin, (double x)) {
+ using FPBits = typename 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^-26.
+ if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 26)) {
+ // When |x| < 2^-26, the relative error of the approximation asin(x) ~ x
+ // is:
+ // |asin(x) - x| / |asin(x)| < |x^3| / (6|x|)
+ // = x^2 / 6
+ // < 2^-54
+ // < epsilon(1)/2.
+ // So the correctly rounded values of asin(x) are:
+ // = x + sign(x)*eps(x) if rounding mode = FE_TOWARDZERO,
+ // or (rounding mode = FE_UPWARD and x is
+ // negative),
+ // = x otherwise.
+ // To simplify the rounding decision and make it more efficient, we use
+ // fma(x, 2^-54, x) instead.
+ // Note: to use the formula x + 2^-54*x to decide the correct rounding, we
+ // do need fma(x, 2^-54, x) to prevent underflow caused by 2^-54*x when
+ // |x| < 2^-1022. For targets without FMA instructions, when x is close to
+ // denormal range, we normalize x,
+#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)
+ return x;
+#elif defined(LIBC_TARGET_CPU_HAS_FMA_DOUBLE)
+ return fputil::multiply_add(x, 0x1.0p-54, x);
+#else
+ if (xbits.abs().uintval() == 0)
+ return x;
+ // Get sign(x) * min_normal.
+ FPBits eps_bits = FPBits::min_normal();
+ eps_bits.set_sign(xbits.sign());
+ double eps = eps_bits.get_val();
+ double normalize_const = (x_exp == 0) ? eps : 0.0;
+ double scaled_normal =
+ fputil::multiply_add(x + normalize_const, 0x1.0p54, eps);
+ return fputil::multiply_add(scaled_normal, 0x1.0p-54, -normalize_const);
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ }
+
+#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ return x * asin_eval(x * x);
+#else
+ unsigned idx;
+ DoubleDouble x_sq = fputil::exact_mult(x, x);
+ double err = x * 0x1.0p-51;
+ // Polynomial approximation:
+ // p ~ asin(x)/x
+
+ DoubleDouble p = asin_eval(x_sq, idx, err);
+ // asin(x) ~ x * (ASIN_COEFFS[idx][0] + p)
+ DoubleDouble r0 = fputil::exact_mult(x, p.hi);
+ double r_lo = fputil::multiply_add(x, p.lo, r0.lo);
+
+ // Ziv's accuracy test.
+
+ double r_upper = r0.hi + (r_lo + err);
+ double r_lower = r0.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/21 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);
+ Float128 r = 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) {
+ // return +- pi/2
+ return fputil::multiply_add(x_sign, PI_OVER_TWO.hi,
+ x_sign * PI_OVER_TWO.lo);
+ }
+ // |x| > 1, return NaN.
+ if (xbits.is_finite()) {
+ 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:
+ //
+ // Assume further that 0.5 < x <= 1, and let:
+ // y = asin(x)
+ // We will use the double angle formula:
+ // cos(2y) = 1 - 2 sin^2(y)
+ // and the complement angle identity:
+ // x = sin(y) = cos(pi/2 - y)
+ // = 1 - 2 sin^2 (pi/4 - y/2)
+ // So:
+ // sin(pi/4 - y/2) = sqrt( (1 - x)/2 )
+ // And hence:
+ // pi/4 - y/2 = asin( sqrt( (1 - x)/2 ) )
+ // Equivalently:
+ // asin(x) = y = pi/2 - 2 * asin( sqrt( (1 - x)/2 ) )
+ // Let u = (1 - x)/2, then:
+ // asin(x) = pi/2 - 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:
+ // asin(x) ~ pi/2 - 2 * sqrt(u) * P(u),
+
+ // 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).
+ // Then,
+ // asin(x) ~ pi/2 - 2*(v_hi + v_lo) * P(u)
+ double v_hi = fputil::sqrt<double>(u);
+
+#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+ double p = asin_eval(u);
+ double r = x_sign * fputil::multiply_add(-2.0 * v_hi, p, PI_OVER_TWO.hi);
+ 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;
+ [[maybe_unused]] double err = vh * 0x1.0p-51;
+
+ DoubleDouble p = asin_eval(DoubleDouble{0.0, u}, idx, err);
----------------
overmighty wrote:
Why `[[maybe_unused]]`?
https://github.com/llvm/llvm-project/pull/134401
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