1 /*
2  * Single-precision pow function.
3  *
4  * Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
5  * See https://llvm.org/LICENSE.txt for license information.
6  * SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
7  */
8 
9 #include <math.h>
10 #include <stdint.h>
11 #include "math_config.h"
12 
13 /*
14 POWF_LOG2_POLY_ORDER = 5
15 EXP2F_TABLE_BITS = 5
16 
17 ULP error: 0.82 (~ 0.5 + relerr*2^24)
18 relerr: 1.27 * 2^-26 (Relative error ~= 128*Ln2*relerr_log2 + relerr_exp2)
19 relerr_log2: 1.83 * 2^-33 (Relative error of logx.)
20 relerr_exp2: 1.69 * 2^-34 (Relative error of exp2(ylogx).)
21 */
22 
23 #define N (1 << POWF_LOG2_TABLE_BITS)
24 #define T __powf_log2_data.tab
25 #define A __powf_log2_data.poly
26 #define OFF 0x3f330000
27 
28 /* Subnormal input is normalized so ix has negative biased exponent.
29    Output is multiplied by N (POWF_SCALE) if TOINT_INTRINSICS is set.  */
30 static inline double_t
log2_inline(uint32_t ix)31 log2_inline (uint32_t ix)
32 {
33   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
34   double_t z, r, r2, r4, p, q, y, y0, invc, logc;
35   uint32_t iz, top, tmp;
36   int k, i;
37 
38   /* x = 2^k z; where z is in range [OFF,2*OFF] and exact.
39      The range is split into N subintervals.
40      The ith subinterval contains z and c is near its center.  */
41   tmp = ix - OFF;
42   i = (tmp >> (23 - POWF_LOG2_TABLE_BITS)) % N;
43   top = tmp & 0xff800000;
44   iz = ix - top;
45   k = (int32_t) top >> (23 - POWF_SCALE_BITS); /* arithmetic shift */
46   invc = T[i].invc;
47   logc = T[i].logc;
48   z = (double_t) asfloat (iz);
49 
50   /* log2(x) = log1p(z/c-1)/ln2 + log2(c) + k */
51   r = z * invc - 1;
52   y0 = logc + (double_t) k;
53 
54   /* Pipelined polynomial evaluation to approximate log1p(r)/ln2.  */
55   r2 = r * r;
56   y = A[0] * r + A[1];
57   p = A[2] * r + A[3];
58   r4 = r2 * r2;
59   q = A[4] * r + y0;
60   q = p * r2 + q;
61   y = y * r4 + q;
62   return y;
63 }
64 
65 #undef N
66 #undef T
67 #define N (1 << EXP2F_TABLE_BITS)
68 #define T __exp2f_data.tab
69 #define SIGN_BIAS (1 << (EXP2F_TABLE_BITS + 11))
70 
71 /* The output of log2 and thus the input of exp2 is either scaled by N
72    (in case of fast toint intrinsics) or not.  The unscaled xd must be
73    in [-1021,1023], sign_bias sets the sign of the result.  */
74 static inline float
exp2_inline(double_t xd,uint32_t sign_bias)75 exp2_inline (double_t xd, uint32_t sign_bias)
76 {
77   uint64_t ki, ski, t;
78   /* double_t for better performance on targets with FLT_EVAL_METHOD==2.  */
79   double_t kd, z, r, r2, y, s;
80 
81 #if TOINT_INTRINSICS
82 # define C __exp2f_data.poly_scaled
83   /* N*x = k + r with r in [-1/2, 1/2] */
84   kd = roundtoint (xd); /* k */
85   ki = converttoint (xd);
86 #else
87 # define C __exp2f_data.poly
88 # define SHIFT __exp2f_data.shift_scaled
89   /* x = k/N + r with r in [-1/(2N), 1/(2N)] */
90   kd = eval_as_double (xd + SHIFT);
91   ki = asuint64 (kd);
92   kd -= SHIFT; /* k/N */
93 #endif
94   r = xd - kd;
95 
96   /* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
97   t = T[ki % N];
98   ski = ki + sign_bias;
99   t += ski << (52 - EXP2F_TABLE_BITS);
100   s = asdouble (t);
101   z = C[0] * r + C[1];
102   r2 = r * r;
103   y = C[2] * r + 1;
104   y = z * r2 + y;
105   y = y * s;
106   return eval_as_float (y);
107 }
108 
109 /* Returns 0 if not int, 1 if odd int, 2 if even int.  The argument is
110    the bit representation of a non-zero finite floating-point value.  */
111 static inline int
checkint(uint32_t iy)112 checkint (uint32_t iy)
113 {
114   int e = iy >> 23 & 0xff;
115   if (e < 0x7f)
116     return 0;
117   if (e > 0x7f + 23)
118     return 2;
119   if (iy & ((1 << (0x7f + 23 - e)) - 1))
120     return 0;
121   if (iy & (1 << (0x7f + 23 - e)))
122     return 1;
123   return 2;
124 }
125 
126 static inline int
zeroinfnan(uint32_t ix)127 zeroinfnan (uint32_t ix)
128 {
129   return 2 * ix - 1 >= 2u * 0x7f800000 - 1;
130 }
131 
132 float
powf(float x,float y)133 powf (float x, float y)
134 {
135   uint32_t sign_bias = 0;
136   uint32_t ix, iy;
137 
138   ix = asuint (x);
139   iy = asuint (y);
140   if (unlikely (ix - 0x00800000 >= 0x7f800000 - 0x00800000 || zeroinfnan (iy)))
141     {
142       /* Either (x < 0x1p-126 or inf or nan) or (y is 0 or inf or nan).  */
143       if (unlikely (zeroinfnan (iy)))
144 	{
145 	  if (2 * iy == 0)
146 	    return issignalingf_inline (x) ? x + y : 1.0f;
147 	  if (ix == 0x3f800000)
148 	    return issignalingf_inline (y) ? x + y : 1.0f;
149 	  if (2 * ix > 2u * 0x7f800000 || 2 * iy > 2u * 0x7f800000)
150 	    return x + y;
151 	  if (2 * ix == 2 * 0x3f800000)
152 	    return 1.0f;
153 	  if ((2 * ix < 2 * 0x3f800000) == !(iy & 0x80000000))
154 	    return 0.0f; /* |x|<1 && y==inf or |x|>1 && y==-inf.  */
155 	  return y * y;
156 	}
157       if (unlikely (zeroinfnan (ix)))
158 	{
159 	  float_t x2 = x * x;
160 	  if (ix & 0x80000000 && checkint (iy) == 1)
161 	    {
162 	      x2 = -x2;
163 	      sign_bias = 1;
164 	    }
165 #if WANT_ERRNO
166 	  if (2 * ix == 0 && iy & 0x80000000)
167 	    return __math_divzerof (sign_bias);
168 #endif
169 	  /* Without the barrier some versions of clang hoist the 1/x2 and
170 	     thus division by zero exception can be signaled spuriously.  */
171 	  return iy & 0x80000000 ? opt_barrier_float (1 / x2) : x2;
172 	}
173       /* x and y are non-zero finite.  */
174       if (ix & 0x80000000)
175 	{
176 	  /* Finite x < 0.  */
177 	  int yint = checkint (iy);
178 	  if (yint == 0)
179 	    return __math_invalidf (x);
180 	  if (yint == 1)
181 	    sign_bias = SIGN_BIAS;
182 	  ix &= 0x7fffffff;
183 	}
184       if (ix < 0x00800000)
185 	{
186 	  /* Normalize subnormal x so exponent becomes negative.  */
187 	  ix = asuint (x * 0x1p23f);
188 	  ix &= 0x7fffffff;
189 	  ix -= 23 << 23;
190 	}
191     }
192   double_t logx = log2_inline (ix);
193   double_t ylogx = y * logx; /* Note: cannot overflow, y is single prec.  */
194   if (unlikely ((asuint64 (ylogx) >> 47 & 0xffff)
195 		 >= asuint64 (126.0 * POWF_SCALE) >> 47))
196     {
197       /* |y*log(x)| >= 126.  */
198       if (ylogx > 0x1.fffffffd1d571p+6 * POWF_SCALE)
199 	/* |x^y| > 0x1.ffffffp127.  */
200 	return __math_oflowf (sign_bias);
201       if (WANT_ROUNDING && WANT_ERRNO
202 	  && ylogx > 0x1.fffffffa3aae2p+6 * POWF_SCALE)
203 	/* |x^y| > 0x1.fffffep127, check if we round away from 0.  */
204 	if ((!sign_bias
205 	     && eval_as_float (1.0f + opt_barrier_float (0x1p-25f)) != 1.0f)
206 	    || (sign_bias
207 		&& eval_as_float (-1.0f - opt_barrier_float (0x1p-25f))
208 		     != -1.0f))
209 	  return __math_oflowf (sign_bias);
210       if (ylogx <= -150.0 * POWF_SCALE)
211 	return __math_uflowf (sign_bias);
212 #if WANT_ERRNO_UFLOW
213       if (ylogx < -149.0 * POWF_SCALE)
214 	return __math_may_uflowf (sign_bias);
215 #endif
216     }
217   return exp2_inline (ylogx, sign_bias);
218 }
219 #if USE_GLIBC_ABI
220 strong_alias (powf, __powf_finite)
221 hidden_alias (powf, __ieee754_powf)
222 #endif
223