1//===-- Implementation header for acos --------------------------*- C++ -*-===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H
11
12#include "asin_utils.h"
13#include "src/__support/FPUtil/FEnvImpl.h"
14#include "src/__support/FPUtil/FPBits.h"
15#include "src/__support/FPUtil/double_double.h"
16#include "src/__support/FPUtil/dyadic_float.h"
17#include "src/__support/FPUtil/multiply_add.h"
18#include "src/__support/FPUtil/sqrt.h"
19#include "src/__support/macros/config.h"
20#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
21#include "src/__support/macros/properties/cpu_features.h" // LIBC_TARGET_CPU_HAS_FMA
22
23namespace LIBC_NAMESPACE_DECL {
24
25namespace math {
26
27LIBC_INLINE double acos(double x) {
28 using DoubleDouble = fputil::DoubleDouble;
29 using namespace asin_internal;
30 using FPBits = fputil::FPBits<double>;
31
32 FPBits xbits(x);
33 int x_exp = xbits.get_biased_exponent();
34
35 // |x| < 0.5.
36 if (x_exp < FPBits::EXP_BIAS - 1) {
37 // |x| < 2^-55.
38 if (LIBC_UNLIKELY(x_exp < FPBits::EXP_BIAS - 55)) {
39 // When |x| < 2^-55, acos(x) = pi/2
40#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS)
41 return PI_OVER_TWO.hi;
42#else
43 // Force the evaluation and prevent constant propagation so that it
44 // is rounded correctly for FE_UPWARD rounding mode.
45 return (xbits.abs().get_val() + 0x1.0p-160) + PI_OVER_TWO.hi;
46#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
47 }
48
49#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
50 // acos(x) = pi/2 - asin(x)
51 // = pi/2 - x * P(x^2)
52 double p = asin_eval(x * x);
53 return PI_OVER_TWO.hi + fputil::multiply_add(-x, p, PI_OVER_TWO.lo);
54#else
55 unsigned idx = 0;
56 DoubleDouble x_sq = fputil::exact_mult(a: x, b: x);
57 double err = xbits.abs().get_val() * 0x1.0p-51;
58 // Polynomial approximation:
59 // p ~ asin(x)/x
60 DoubleDouble p = asin_eval(u: x_sq, idx, err);
61 // asin(x) ~ x * p
62 DoubleDouble r0 = fputil::exact_mult(a: x, b: p.hi);
63 // acos(x) = pi/2 - asin(x)
64 // ~ pi/2 - x * p
65 // = pi/2 - x * (p.hi + p.lo)
66 double r_hi = fputil::multiply_add(x: -x, y: p.hi, z: PI_OVER_TWO.hi);
67 // Use Dekker's 2SUM algorithm to compute the lower part.
68 double r_lo = ((PI_OVER_TWO.hi - r_hi) - r0.hi) - r0.lo;
69 r_lo = fputil::multiply_add(x: -x, y: p.lo, z: r_lo + PI_OVER_TWO.lo);
70
71 // Ziv's accuracy test.
72
73 double r_upper = r_hi + (r_lo + err);
74 double r_lower = r_hi + (r_lo - err);
75
76 if (LIBC_LIKELY(r_upper == r_lower))
77 return r_upper;
78
79 // Ziv's accuracy test failed, perform 128-bit calculation.
80
81 // Recalculate mod 1/64.
82 idx = static_cast<unsigned>(fputil::nearest_integer(x: x_sq.hi * 0x1.0p6));
83
84 // Get x^2 - idx/64 exactly. When FMA is available, double-double
85 // multiplication will be correct for all rounding modes. Otherwise we use
86 // DFloat128 directly.
87 DFloat128 x_f128(x);
88
89#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
90 // u = x^2 - idx/64
91 DFloat128 u_hi(
92 fputil::multiply_add(static_cast<double>(idx), -0x1.0p-6, x_sq.hi));
93 DFloat128 u = fputil::quick_add(u_hi, DFloat128(x_sq.lo));
94#else
95 DFloat128 x_sq_f128 = fputil::quick_mul(a: x_f128, b: x_f128);
96 DFloat128 u = fputil::quick_add(
97 a: x_sq_f128, b: DFloat128(static_cast<double>(idx) * (-0x1.0p-6)));
98#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
99
100 DFloat128 p_f128 = asin_eval(u, idx);
101 // Flip the sign of x_f128 to perform subtraction.
102 x_f128.sign = x_f128.sign.negate();
103 DFloat128 r =
104 fputil::quick_add(a: PI_OVER_TWO_F128, b: fputil::quick_mul(a: x_f128, b: p_f128));
105
106 return static_cast<double>(r);
107#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
108 }
109 // |x| >= 0.5
110
111 double x_abs = xbits.abs().get_val();
112
113 // Maintaining the sign:
114 constexpr double SIGN[2] = {1.0, -1.0};
115 double x_sign = SIGN[xbits.is_neg()];
116 // |x| >= 1
117 if (LIBC_UNLIKELY(x_exp >= FPBits::EXP_BIAS)) {
118 // x = +-1, asin(x) = +- pi/2
119 if (x_abs == 1.0) {
120 // x = 1, acos(x) = 0,
121 // x = -1, acos(x) = pi
122 return x == 1.0 ? 0.0 : fputil::multiply_add(x: -x_sign, y: PI.hi, z: PI.lo);
123 }
124 // |x| > 1, return NaN.
125 if (xbits.is_quiet_nan())
126 return x;
127
128 // Set domain error for non-NaN input.
129 if (!xbits.is_nan())
130 fputil::set_errno_if_required(EDOM);
131
132 fputil::raise_except_if_required(FE_INVALID);
133 return FPBits::quiet_nan().get_val();
134 }
135
136 // When |x| >= 0.5, we perform range reduction as follow:
137 //
138 // When 0.5 <= x < 1, let:
139 // y = acos(x)
140 // We will use the double angle formula:
141 // cos(2y) = 1 - 2 sin^2(y)
142 // and the complement angle identity:
143 // x = cos(y) = 1 - 2 sin^2 (y/2)
144 // So:
145 // sin(y/2) = sqrt( (1 - x)/2 )
146 // And hence:
147 // y/2 = asin( sqrt( (1 - x)/2 ) )
148 // Equivalently:
149 // acos(x) = y = 2 * asin( sqrt( (1 - x)/2 ) )
150 // Let u = (1 - x)/2, then:
151 // acos(x) = 2 * asin( sqrt(u) )
152 // Moreover, since 0.5 <= x < 1:
153 // 0 < u <= 1/4, and 0 < sqrt(u) <= 0.5,
154 // And hence we can reuse the same polynomial approximation of asin(x) when
155 // |x| <= 0.5:
156 // acos(x) ~ 2 * sqrt(u) * P(u).
157 //
158 // When -1 < x <= -0.5, we reduce to the previous case using the formula:
159 // acos(x) = pi - acos(-x)
160 // = pi - 2 * asin ( sqrt( (1 + x)/2 ) )
161 // ~ pi - 2 * sqrt(u) * P(u),
162 // where u = (1 - |x|)/2.
163
164 // u = (1 - |x|)/2
165 double u = fputil::multiply_add(x: x_abs, y: -0.5, z: 0.5);
166 // v_hi + v_lo ~ sqrt(u).
167 // Let:
168 // h = u - v_hi^2 = (sqrt(u) - v_hi) * (sqrt(u) + v_hi)
169 // Then:
170 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
171 // ~ v_hi + h / (2 * v_hi)
172 // So we can use:
173 // v_lo = h / (2 * v_hi).
174 double v_hi = fputil::sqrt<double>(x: u);
175
176#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
177 constexpr DoubleDouble CONST_TERM[2] = {{0.0, 0.0}, PI};
178 DoubleDouble const_term = CONST_TERM[xbits.is_neg()];
179
180 double p = asin_eval(u);
181 double scale = x_sign * 2.0 * v_hi;
182 double r = const_term.hi + fputil::multiply_add(scale, p, const_term.lo);
183 return r;
184#else
185
186#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
187 double h = fputil::multiply_add(v_hi, -v_hi, u);
188#else
189 DoubleDouble v_hi_sq = fputil::exact_mult(a: v_hi, b: v_hi);
190 double h = (u - v_hi_sq.hi) - v_hi_sq.lo;
191#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
192
193 // Scale v_lo and v_hi by 2 from the formula:
194 // vh = v_hi * 2
195 // vl = 2*v_lo = h / v_hi.
196 double vh = v_hi * 2.0;
197 double vl = h / v_hi;
198
199 // Polynomial approximation:
200 // p ~ asin(sqrt(u))/sqrt(u)
201 unsigned idx = 0;
202 double err = vh * 0x1.0p-51;
203
204 DoubleDouble p = asin_eval(u: DoubleDouble{.lo: 0.0, .hi: u}, idx, err);
205
206 // Perform computations in double-double arithmetic:
207 // asin(x) = pi/2 - (v_hi + v_lo) * (ASIN_COEFFS[idx][0] + p)
208 DoubleDouble r0 = fputil::quick_mult(a: DoubleDouble{.lo: vl, .hi: vh}, b: p);
209
210 double r_hi = 0, r_lo = 0;
211 if (xbits.is_pos()) {
212 r_hi = r0.hi;
213 r_lo = r0.lo;
214 } else {
215 DoubleDouble r = fputil::exact_add(a: PI.hi, b: -r0.hi);
216 r_hi = r.hi;
217 r_lo = (PI.lo - r0.lo) + r.lo;
218 }
219
220 // Ziv's accuracy test.
221
222 double r_upper = r_hi + (r_lo + err);
223 double r_lower = r_hi + (r_lo - err);
224
225 if (LIBC_LIKELY(r_upper == r_lower))
226 return r_upper;
227
228 // Ziv's accuracy test failed, we redo the computations in DFloat128.
229 // Recalculate mod 1/64.
230 idx = static_cast<unsigned>(fputil::nearest_integer(x: u * 0x1.0p6));
231
232 // After the first step of Newton-Raphson approximating v = sqrt(u), we have
233 // that:
234 // sqrt(u) = v_hi + h / (sqrt(u) + v_hi)
235 // v_lo = h / (2 * v_hi)
236 // With error:
237 // sqrt(u) - (v_hi + v_lo) = h * ( 1/(sqrt(u) + v_hi) - 1/(2*v_hi) )
238 // = -h^2 / (2*v * (sqrt(u) + v)^2).
239 // Since:
240 // (sqrt(u) + v_hi)^2 ~ (2sqrt(u))^2 = 4u,
241 // we can add another correction term to (v_hi + v_lo) that is:
242 // v_ll = -h^2 / (2*v_hi * 4u)
243 // = -v_lo * (h / 4u)
244 // = -vl * (h / 8u),
245 // making the errors:
246 // sqrt(u) - (v_hi + v_lo + v_ll) = O(h^3)
247 // well beyond 128-bit precision needed.
248
249 // Get the rounding error of vl = 2 * v_lo ~ h / vh
250 // Get full product of vh * vl
251#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
252 double vl_lo = fputil::multiply_add(-v_hi, vl, h) / v_hi;
253#else
254 DoubleDouble vh_vl = fputil::exact_mult(a: v_hi, b: vl);
255 double vl_lo = ((h - vh_vl.hi) - vh_vl.lo) / v_hi;
256#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE
257 // vll = 2*v_ll = -vl * (h / (4u)).
258 double t = h * (-0.25) / u;
259 double vll = fputil::multiply_add(x: vl, y: t, z: vl_lo);
260 // m_v = -(v_hi + v_lo + v_ll).
261 DFloat128 m_v = fputil::quick_add(
262 a: DFloat128(vh), b: fputil::quick_add(a: DFloat128(vl), b: DFloat128(vll)));
263 m_v.sign = xbits.sign();
264
265 // Perform computations in DFloat128:
266 // acos(x) = (v_hi + v_lo + vll) * P(u) , when 0.5 <= x < 1,
267 // = pi - (v_hi + v_lo + vll) * P(u) , when -1 < x <= -0.5.
268 DFloat128 y_f128(
269 fputil::multiply_add(x: static_cast<double>(idx), y: -0x1.0p-6, z: u));
270
271 DFloat128 p_f128 = asin_eval(u: y_f128, idx);
272 DFloat128 r_f128 = fputil::quick_mul(a: m_v, b: p_f128);
273
274 if (xbits.is_neg())
275 r_f128 = fputil::quick_add(a: PI_F128, b: r_f128);
276
277 return static_cast<double>(r_f128);
278#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
279}
280
281} // namespace math
282
283} // namespace LIBC_NAMESPACE_DECL
284
285#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ACOS_H
286