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-rw-r--r--libc/src/math/generic/acos.cpp266
1 files changed, 2 insertions, 264 deletions
diff --git a/libc/src/math/generic/acos.cpp b/libc/src/math/generic/acos.cpp
index c14721f..3a59642 100644
--- a/libc/src/math/generic/acos.cpp
+++ b/libc/src/math/generic/acos.cpp
@@ -7,272 +7,10 @@
//===----------------------------------------------------------------------===//
#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
+#include "src/__support/math/acos.h"
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
-}
+LLVM_LIBC_FUNCTION(double, acos, (double x)) { return math::acos(x); }
} // namespace LIBC_NAMESPACE_DECL
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