[libc][math] Refactor acosf implementation to header-only in src/__support/math folder.

libc/shared/math.h

@@ -12,6 +12,7 @@
 #include "libc_common.h"
 
 #include "math/acos.h"
+#include "math/acosf.h"
 #include "math/exp.h"
 #include "math/exp10.h"
 #include "math/exp10f.h"

libc/shared/math/acosf.h

@@ -0,0 +1,23 @@
+//===-- Shared acosf function -----------------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_ACOSF_H
+#define LLVM_LIBC_SHARED_MATH_ACOSF_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/acosf.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::acosf;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_ACOSF_H

libc/src/__support/math/CMakeLists.txt

@@ -17,6 +17,20 @@ add_header_library(
     libc.src.__support.macros.properties.cpu_features
 )
 
+add_header_library(
+  acosf
+  HDRS
+    acosf.h
+  DEPENDS
+    .inv_trigf_utils
+    libc.src.__support.FPUtil.except_value_utils
+    libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.polyeval
+    libc.src.__support.FPUtil.sqrt
+    libc.src.__support.macros.optimization
+)
+
 add_header_library(
   asin_utils
   HDRS
@@ -98,6 +112,16 @@ add_header_library(
     libc.src.__support.FPUtil.manipulation_functions
 )
 
+add_header_library(
+  inv_trigf_utils
+  HDRS
+    inv_trigf_utils.h
+  DEPENDS
+    libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.polyeval
+    libc.src.__support.common
+)
+
 add_header_library(
   frexpf16
   HDRS

libc/src/__support/math/acosf.h

@@ -0,0 +1,139 @@
+//===-- Implementation header for acosf -------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ACOSF_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_ACOSF_H
+
+#include "inv_trigf_utils.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/except_value_utils.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
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace math {
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+static constexpr size_t N_EXCEPTS = 4;
+
+// Exceptional values when |x| <= 0.5
+static constexpr fputil::ExceptValues<float, N_EXCEPTS> ACOSF_EXCEPTS = {{
+    // (inputs, RZ output, RU offset, RD offset, RN offset)
+    // x = 0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ)
+    {0x328885a3, 0x3fc90fda, 1, 0, 1},
+    // x = -0x1.110b46p-26, acosf(x) = 0x1.921fb4p0 (RZ)
+    {0xb28885a3, 0x3fc90fda, 1, 0, 1},
+    // x = 0x1.04c444p-12, acosf(x) = 0x1.920f68p0 (RZ)
+    {0x39826222, 0x3fc907b4, 1, 0, 1},
+    // x = -0x1.04c444p-12, acosf(x) = 0x1.923p0 (RZ)
+    {0xb9826222, 0x3fc91800, 1, 0, 1},
+}};
+#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+static constexpr float acosf(float x) {
+  using FPBits = typename fputil::FPBits<float>;
+
+  FPBits xbits(x);
+  uint32_t x_uint = xbits.uintval();
+  uint32_t x_abs = xbits.uintval() & 0x7fff'ffffU;
+  uint32_t x_sign = x_uint >> 31;
+
+  // |x| <= 0.5
+  if (LIBC_UNLIKELY(x_abs <= 0x3f00'0000U)) {
+    // |x| < 0x1p-10
+    if (LIBC_UNLIKELY(x_abs < 0x3a80'0000U)) {
+      // When |x| < 2^-10, we use the following approximation:
+      //   acos(x) = pi/2 - asin(x)
+      //           ~ pi/2 - x - x^3 / 6
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+      // Check for exceptional values
+      if (auto r = ACOSF_EXCEPTS.lookup(x_uint); LIBC_UNLIKELY(r.has_value()))
+        return r.value();
+#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+
+      double xd = static_cast<double>(x);
+      return static_cast<float>(fputil::multiply_add(
+          -0x1.5555555555555p-3 * xd, xd * xd, M_MATH_PI_2 - xd));
+    }
+
+    // For |x| <= 0.5, we approximate acosf(x) by:
+    //   acos(x) = pi/2 - asin(x) = pi/2 - x * P(x^2)
+    // Where P(X^2) = Q(X) is a degree-20 minimax even polynomial approximating
+    // asin(x)/x on [0, 0.5] generated by Sollya with:
+    // > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|],
+    //                 [|1, D...|], [0, 0.5]);
+    double xd = static_cast<double>(x);
+    double xsq = xd * xd;
+    double x3 = xd * xsq;
+    double r = asin_eval(xsq);
+    return static_cast<float>(fputil::multiply_add(-x3, r, M_MATH_PI_2 - xd));
+  }
+
+  // |x| >= 1, return 0, 2pi, or NaNs.
+  if (LIBC_UNLIKELY(x_abs >= 0x3f80'0000U)) {
+    if (x_abs == 0x3f80'0000U)
+      return x_sign ? /* x == -1.0f */ fputil::round_result_slightly_down(
+                          0x1.921fb6p+1f)
+                    : /* x == 1.0f */ 0.0f;
+
+    if (xbits.is_signaling_nan()) {
+      fputil::raise_except_if_required(FE_INVALID);
+      return FPBits::quiet_nan().get_val();
+    }
+
+    // |x| <= +/-inf
+    if (x_abs <= 0x7f80'0000U) {
+      fputil::set_errno_if_required(EDOM);
+      fputil::raise_except_if_required(FE_INVALID);
+    }
+
+    return x + FPBits::quiet_nan().get_val();
+  }
+
+  // When 0.5 < |x| < 1, we perform range reduction as follow:
+  //
+  // Assume further that 0.5 < x <= 1, and let:
+  //   y = acos(x)
+  // We use the double angle formula:
+  //   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 ) )
+  // 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 use the identity:
+  //   acos(x) = pi - acos(-x)
+  // which is reduced to the postive case.
+
+  xbits.set_sign(Sign::POS);
+  double xd = static_cast<double>(xbits.get_val());
+  double u = fputil::multiply_add(-0.5, xd, 0.5);
+  double cv = 2 * fputil::sqrt<double>(u);
+
+  double r3 = asin_eval(u);
+  double r = fputil::multiply_add(cv * u, r3, cv);
+  return static_cast<float>(x_sign ? M_MATH_PI - r : r);
+}
+
+} // namespace math
+
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H

libc/src/math/generic/inv_trigf_utils.cpp → libc/src/__support/math/inv_trigf_utils.h

@@ -1,16 +1,25 @@
-//===-- Single-precision general exp/log functions ------------------------===//
+//===-- Single-precision general inverse trigonometric functions ----------===//
 //
 // 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 "inv_trigf_utils.h"
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_INV_TRIGF_UTILS_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_INV_TRIGF_UTILS_H
+
+#include "src/__support/FPUtil/PolyEval.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/common.h"
 #include "src/__support/macros/config.h"
 
 namespace LIBC_NAMESPACE_DECL {
 
+// PI and PI / 2
+static constexpr double M_MATH_PI = 0x1.921fb54442d18p+1;
+static constexpr double M_MATH_PI_2 = 0x1.921fb54442d18p+0;
+
 // Polynomial approximation for 0 <= x <= 1:
 //   atan(x) ~ atan((i/16) + (x - (i/16)) * Q(x - i/16)
 //           = P(x - i/16)
@@ -29,7 +38,7 @@ namespace LIBC_NAMESPACE_DECL {
 // Notice that degree-7 is good enough for atanf, but degree-8 helps reduce the
 // error bounds for atan2f's fast pass 16 times, and it does not affect the
 // performance of atanf much.
-double ATAN_COEFFS[17][9] = {
+static constexpr double ATAN_COEFFS[17][9] = {
     {0.0, 1.0, 0x1.3f8d76d26d61bp-47, -0x1.5555555574cd8p-2,
      0x1.0dde5d06878eap-29, 0x1.99997738acc77p-3, 0x1.2c43eac9797cap-16,
      -0x1.25fb020007dbdp-3, 0x1.c1b6c31d7b0aep-7},
@@ -83,4 +92,89 @@ double ATAN_COEFFS[17][9] = {
      0x1.555e31a1e15e9p-6, -0x1.245240d65e629p-7, -0x1.fa9ba66478903p-11},
 };
 
+// Look-up table for atan(k/16) with k = 0..16.
+static constexpr double ATAN_K_OVER_16[17] = {
+    0.0,
+    0x1.ff55bb72cfdeap-5,
+    0x1.fd5ba9aac2f6ep-4,
+    0x1.7b97b4bce5b02p-3,
+    0x1.f5b75f92c80ddp-3,
+    0x1.362773707ebccp-2,
+    0x1.6f61941e4def1p-2,
+    0x1.a64eec3cc23fdp-2,
+    0x1.dac670561bb4fp-2,
+    0x1.0657e94db30dp-1,
+    0x1.1e00babdefeb4p-1,
+    0x1.345f01cce37bbp-1,
+    0x1.4978fa3269ee1p-1,
+    0x1.5d58987169b18p-1,
+    0x1.700a7c5784634p-1,
+    0x1.819d0b7158a4dp-1,
+    0x1.921fb54442d18p-1,
+};
+
+// For |x| <= 1/32 and 0 <= i <= 16, return Q(x) such that:
+//   Q(x) ~ (atan(x + i/16) - atan(i/16)) / x.
+LIBC_INLINE static double atan_eval(double x, unsigned i) {
+  double x2 = x * x;
+
+  double c0 = fputil::multiply_add(x, ATAN_COEFFS[i][2], ATAN_COEFFS[i][1]);
+  double c1 = fputil::multiply_add(x, ATAN_COEFFS[i][4], ATAN_COEFFS[i][3]);
+  double c2 = fputil::multiply_add(x, ATAN_COEFFS[i][6], ATAN_COEFFS[i][5]);
+  double c3 = fputil::multiply_add(x, ATAN_COEFFS[i][8], ATAN_COEFFS[i][7]);
+
+  double x4 = x2 * x2;
+  double d1 = fputil::multiply_add(x2, c1, c0);
+  double d2 = fputil::multiply_add(x2, c3, c2);
+  double p = fputil::multiply_add(x4, d2, d1);
+  return p;
+}
+
+// Evaluate atan without big lookup table.
+//   atan(n/d) - atan(k/16) = atan((n/d - k/16) / (1 + (n/d) * (k/16)))
+//                          = atan((n - d * k/16)) / (d + n * k/16))
+// So we let q = (n - d * k/16) / (d + n * k/16),
+// and approximate with Taylor polynomial:
+//   atan(q) ~ q - q^3/3 + q^5/5 - q^7/7 + q^9/9
+LIBC_INLINE static double atan_eval_no_table(double num, double den,
+                                             double k_over_16) {
+  double num_r = fputil::multiply_add(den, -k_over_16, num);
+  double den_r = fputil::multiply_add(num, k_over_16, den);
+  double q = num_r / den_r;
+
+  constexpr double ATAN_TAYLOR[] = {
+      -0x1.5555555555555p-2,
+      0x1.999999999999ap-3,
+      -0x1.2492492492492p-3,
+      0x1.c71c71c71c71cp-4,
+  };
+  double q2 = q * q;
+  double q3 = q2 * q;
+  double q4 = q2 * q2;
+  double c0 = fputil::multiply_add(q2, ATAN_TAYLOR[1], ATAN_TAYLOR[0]);
+  double c1 = fputil::multiply_add(q2, ATAN_TAYLOR[3], ATAN_TAYLOR[2]);
+  double d = fputil::multiply_add(q4, c1, c0);
+  return fputil::multiply_add(q3, d, q);
+}
+
+// > Q = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20|],
+//                 [|1, D...|], [0, 0.5]);
+static constexpr double ASIN_COEFFS[10] = {
+    0x1.5555555540fa1p-3, 0x1.333333512edc2p-4, 0x1.6db6cc1541b31p-5,
+    0x1.f1caff324770ep-6, 0x1.6e43899f5f4f4p-6, 0x1.1f847cf652577p-6,
+    0x1.9b60f47f87146p-7, 0x1.259e2634c494fp-6, -0x1.df946fa875ddp-8,
+    0x1.02311ecf99c28p-5};
+
+// Evaluate P(x^2) - 1, where P(x^2) ~ asin(x)/x
+LIBC_INLINE static double asin_eval(double xsq) {
+  double x4 = xsq * xsq;
+  double r1 = fputil::polyeval(x4, ASIN_COEFFS[0], ASIN_COEFFS[2],
+                               ASIN_COEFFS[4], ASIN_COEFFS[6], ASIN_COEFFS[8]);
+  double r2 = fputil::polyeval(x4, ASIN_COEFFS[1], ASIN_COEFFS[3],
+                               ASIN_COEFFS[5], ASIN_COEFFS[7], ASIN_COEFFS[9]);
+  return fputil::multiply_add(xsq, r2, r1);
+}
+
 } // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SRC___SUPPORT_MATH_INV_TRIGF_UTILS_H