1 /*
2 * Single-precision scalar tan(x) function.
3 *
4 * Copyright (c) 2021-2023, Arm Limited.
5 * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
6 */
7 #include "math_config.h"
8 #include "pl_sig.h"
9 #include "pl_test.h"
10 #include "pairwise_hornerf.h"
11
12 /* Useful constants. */
13 #define NegPio2_1 (-0x1.921fb6p+0f)
14 #define NegPio2_2 (0x1.777a5cp-25f)
15 #define NegPio2_3 (0x1.ee59dap-50f)
16 /* Reduced from 0x1p20 to 0x1p17 to ensure 3.5ulps. */
17 #define RangeVal (0x1p17f)
18 #define InvPio2 ((0x1.45f306p-1f))
19 #define Shift (0x1.8p+23f)
20 #define AbsMask (0x7fffffff)
21 #define Pio4 (0x1.921fb6p-1)
22 /* 2PI * 2^-64. */
23 #define Pio2p63 (0x1.921FB54442D18p-62)
24
25 #define P(i) __tanf_poly_data.poly_tan[i]
26 #define Q(i) __tanf_poly_data.poly_cotan[i]
27
28 static inline float
eval_P(float z)29 eval_P (float z)
30 {
31 return PAIRWISE_HORNER_5 (z, z * z, P);
32 }
33
34 static inline float
eval_Q(float z)35 eval_Q (float z)
36 {
37 return PAIRWISE_HORNER_3 (z, z * z, Q);
38 }
39
40 /* Reduction of the input argument x using Cody-Waite approach, such that x = r
41 + n * pi/2 with r lives in [-pi/4, pi/4] and n is a signed integer. */
42 static inline float
reduce(float x,int32_t * in)43 reduce (float x, int32_t *in)
44 {
45 /* n = rint(x/(pi/2)). */
46 float r = x;
47 float q = fmaf (InvPio2, r, Shift);
48 float n = q - Shift;
49 /* There is no rounding here, n is representable by a signed integer. */
50 *in = (int32_t) n;
51 /* r = x - n * (pi/2) (range reduction into -pi/4 .. pi/4). */
52 r = fmaf (NegPio2_1, n, r);
53 r = fmaf (NegPio2_2, n, r);
54 r = fmaf (NegPio2_3, n, r);
55 return r;
56 }
57
58 /* Table with 4/PI to 192 bit precision. To avoid unaligned accesses
59 only 8 new bits are added per entry, making the table 4 times larger. */
60 static const uint32_t __inv_pio4[24]
61 = {0x000000a2, 0x0000a2f9, 0x00a2f983, 0xa2f9836e, 0xf9836e4e, 0x836e4e44,
62 0x6e4e4415, 0x4e441529, 0x441529fc, 0x1529fc27, 0x29fc2757, 0xfc2757d1,
63 0x2757d1f5, 0x57d1f534, 0xd1f534dd, 0xf534ddc0, 0x34ddc0db, 0xddc0db62,
64 0xc0db6295, 0xdb629599, 0x6295993c, 0x95993c43, 0x993c4390, 0x3c439041};
65
66 /* Reduce the range of XI to a multiple of PI/2 using fast integer arithmetic.
67 XI is a reinterpreted float and must be >= 2.0f (the sign bit is ignored).
68 Return the modulo between -PI/4 and PI/4 and store the quadrant in NP.
69 Reduction uses a table of 4/PI with 192 bits of precision. A 32x96->128 bit
70 multiply computes the exact 2.62-bit fixed-point modulo. Since the result
71 can have at most 29 leading zeros after the binary point, the double
72 precision result is accurate to 33 bits. */
73 static inline double
reduce_large(uint32_t xi,int * np)74 reduce_large (uint32_t xi, int *np)
75 {
76 const uint32_t *arr = &__inv_pio4[(xi >> 26) & 15];
77 int shift = (xi >> 23) & 7;
78 uint64_t n, res0, res1, res2;
79
80 xi = (xi & 0xffffff) | 0x800000;
81 xi <<= shift;
82
83 res0 = xi * arr[0];
84 res1 = (uint64_t) xi * arr[4];
85 res2 = (uint64_t) xi * arr[8];
86 res0 = (res2 >> 32) | (res0 << 32);
87 res0 += res1;
88
89 n = (res0 + (1ULL << 61)) >> 62;
90 res0 -= n << 62;
91 double x = (int64_t) res0;
92 *np = n;
93 return x * Pio2p63;
94 }
95
96 /* Top 12 bits of the float representation with the sign bit cleared. */
97 static inline uint32_t
top12(float x)98 top12 (float x)
99 {
100 return (asuint (x) >> 20);
101 }
102
103 /* Fast single-precision tan implementation.
104 Maximum ULP error: 3.293ulps.
105 tanf(0x1.c849eap+16) got -0x1.fe8d98p-1 want -0x1.fe8d9ep-1. */
106 float
tanf(float x)107 tanf (float x)
108 {
109 /* Get top words. */
110 uint32_t ix = asuint (x);
111 uint32_t ia = ix & AbsMask;
112 uint32_t ia12 = ia >> 20;
113
114 /* Dispatch between no reduction (small numbers), fast reduction and
115 slow large numbers reduction. The reduction step determines r float
116 (|r| < pi/4) and n signed integer such that x = r + n * pi/2. */
117 int32_t n;
118 float r;
119 if (ia12 < top12 (Pio4))
120 {
121 /* Optimize small values. */
122 if (unlikely (ia12 < top12 (0x1p-12f)))
123 {
124 if (unlikely (ia12 < top12 (0x1p-126f)))
125 /* Force underflow for tiny x. */
126 force_eval_float (x * x);
127 return x;
128 }
129
130 /* tan (x) ~= x + x^3 * P(x^2). */
131 float x2 = x * x;
132 float y = eval_P (x2);
133 return fmaf (x2, x * y, x);
134 }
135 /* Similar to other trigonometric routines, fast inaccurate reduction is
136 performed for values of x from pi/4 up to RangeVal. In order to keep errors
137 below 3.5ulps, we set the value of RangeVal to 2^17. This might differ for
138 other trigonometric routines. Above this value more advanced but slower
139 reduction techniques need to be implemented to reach a similar accuracy.
140 */
141 else if (ia12 < top12 (RangeVal))
142 {
143 /* Fast inaccurate reduction. */
144 r = reduce (x, &n);
145 }
146 else if (ia12 < 0x7f8)
147 {
148 /* Slow accurate reduction. */
149 uint32_t sign = ix & ~AbsMask;
150 double dar = reduce_large (ia, &n);
151 float ar = (float) dar;
152 r = asfloat (asuint (ar) ^ sign);
153 }
154 else
155 {
156 /* tan(Inf or NaN) is NaN. */
157 return __math_invalidf (x);
158 }
159
160 /* If x lives in an interval where |tan(x)|
161 - is finite then use an approximation of tangent in the form
162 tan(r) ~ r + r^3 * P(r^2) = r + r * r^2 * P(r^2).
163 - grows to infinity then use an approximation of cotangent in the form
164 cotan(z) ~ 1/z + z * Q(z^2), where the reciprocal can be computed early.
165 Using symmetries of tangent and the identity tan(r) = cotan(pi/2 - r),
166 we only need to change the sign of r to obtain tan(x) from cotan(r).
167 This 2-interval approach requires 2 different sets of coefficients P and
168 Q, where Q is a lower order polynomial than P. */
169
170 /* Determine if x lives in an interval where |tan(x)| grows to infinity. */
171 uint32_t alt = (uint32_t) n & 1;
172
173 /* Perform additional reduction if required. */
174 float z = alt ? -r : r;
175
176 /* Prepare backward transformation. */
177 float z2 = r * r;
178 float offset = alt ? 1.0f / z : z;
179 float scale = alt ? z : z * z2;
180
181 /* Evaluate polynomial approximation of tan or cotan. */
182 float p = alt ? eval_Q (z2) : eval_P (z2);
183
184 /* A unified way of assembling the result on both interval types. */
185 return fmaf (scale, p, offset);
186 }
187
188 PL_SIG (S, F, 1, tan, -3.1, 3.1)
189 PL_TEST_ULP (tanf, 2.80)
190 PL_TEST_INTERVAL (tanf, 0, 0xffff0000, 10000)
191 PL_TEST_INTERVAL (tanf, 0x1p-127, 0x1p-14, 50000)
192 PL_TEST_INTERVAL (tanf, -0x1p-127, -0x1p-14, 50000)
193 PL_TEST_INTERVAL (tanf, 0x1p-14, 0.7, 50000)
194 PL_TEST_INTERVAL (tanf, -0x1p-14, -0.7, 50000)
195 PL_TEST_INTERVAL (tanf, 0.7, 1.5, 50000)
196 PL_TEST_INTERVAL (tanf, -0.7, -1.5, 50000)
197 PL_TEST_INTERVAL (tanf, 1.5, 0x1p17, 50000)
198 PL_TEST_INTERVAL (tanf, -1.5, -0x1p17, 50000)
199 PL_TEST_INTERVAL (tanf, 0x1p17, 0x1p54, 50000)
200 PL_TEST_INTERVAL (tanf, -0x1p17, -0x1p54, 50000)
201 PL_TEST_INTERVAL (tanf, 0x1p54, inf, 50000)
202 PL_TEST_INTERVAL (tanf, -0x1p54, -inf, 50000)
203