[libc][math] Implement double precision asin correctly rounded for all rounding modes. (#134401)

Main algorithm:

The Taylor series expansion of `asin(x)` is:
```math
\begin{align*}
  asin(x) &= x + x^3 / 6 + 3x^5 / 40 + ... \\
       &= x \cdot P(x^2) \\
       &= x \cdot P(u) &\text{, where } u = x^2.
\end{align*}
```
For the fast path, we perform range reduction mod 1/64 and use degree-7
(minimax + Taylor) polynomials to approximate `P(x^2)`.

When `|x| >= 0.5`, we use the transformation:
```math
  u = \frac{1 + x}{2}
```
and apply half-angle formula to reduce `asin(x)` to:
```math
\begin{align*}
  asin(x) &= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot asin(\sqrt{u}) \right) \\
       &= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot \sqrt{u} \cdot P(u) \right).
\end{align*}
```
Since `0.5 <= |x| <= 1`, `|u| <= 0.5`. So we can reuse the polynomial
evaluation of `P(u)` when `|x| < 0.5`.

For the accurate path, we redo the computations in 128-bit precision
with degree-15 (minimax + Taylor) polynomials to approximate `P(u)`.

1 files changed, 286 insertions, 0 deletions

diff --git a/libc/src/math/generic/asin.cpp b/libc/src/math/generic/asin.cpp
new file mode 100644
index 000000000000..ee85cb1988bb
--- /dev/null
+++ b/libc/src/math/generic/asin.cpp
@@ -0,0 +1,286 @@
+//===-- 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 = 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/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);
+    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);
+    } else if (xbits.is_signaling_nan()) {
+      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;
+  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);
+  DoubleDouble r = fputil::exact_add(PI_OVER_TWO.hi, -r0.hi);
+
+  double r_lo = PI_OVER_TWO.lo - r0.lo + r.lo;
+
+  // Ziv's accuracy test.
+
+#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+  double r_upper = fputil::multiply_add(
+      r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, err));
+  double r_lower = fputil::multiply_add(
+      r.hi, x_sign, fputil::multiply_add(r_lo, x_sign, -err));
+#else
+  r_lo *= x_sign;
+  r.hi *= x_sign;
+  double r_upper = r.hi + (r_lo + err);
+  double r_lower = r.hi + (r_lo - err);
+#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
+
+  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 = Sign::NEG;
+
+  // Perform computations in Float128:
+  //   asin(x) = pi/2 - (v_hi + v_lo + vll) * P(u).
+  Float128 y_f128(fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, u));
+
+  Float128 p_f128 = asin_eval(y_f128, idx);
+  Float128 r0_f128 = fputil::quick_mul(m_v, p_f128);
+  Float128 r_f128 = fputil::quick_add(PI_OVER_TWO_F128, r0_f128);
+
+  if (xbits.is_neg())
+    r_f128.sign = Sign::NEG;
+
+  return static_cast<double>(r_f128);
+#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+}
+
+} // namespace LIBC_NAMESPACE_DECL