[libc][math] Fix parallel implementation for asin (#178100)

Author: bassiounix

libc/src/__support/math/asin.h

@@ -25,7 +25,7 @@ namespace LIBC_NAMESPACE_DECL {
 
 namespace math {
 
-LIBC_INLINE static constexpr double asin(double x) {
+LIBC_INLINE static double asin(double x) {
   using namespace asin_internal;
   using FPBits = fputil::FPBits<double>;
 

libc/src/math/generic/asin.cpp

@@ -11,266 +11,6 @@
 
 namespace LIBC_NAMESPACE_DECL {
 
-LLVM_LIBC_FUNCTION(double, asin, (double x)) {
-  using namespace asin_internal;
-  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 = xbits.abs().get_val() * 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_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:
-  //
-  // 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
-}
+LLVM_LIBC_FUNCTION(double, asin, (double x)) { return math::asin(x); }
 
 } // namespace LIBC_NAMESPACE_DECL