2  * Double-precision 2^x function.
4  * Copyright (c) 2018-2024, Arm Limited.
5  * SPDX-License-Identifier: MIT OR Apache-2.0 WITH LLVM-exception
18 #define C1 __exp_data.exp2_poly[0]
37   if ((ki & 0x80000000) == 0)  in specialcase()
39       /* k > 0, the exponent of scale might have overflowed by 1.  */  in specialcase()
40       sbits -= 1ull << 52;  in specialcase()
45   /* k < 0, need special care in the subnormal range.  */  in specialcase()
52 	 range to avoid double rounding that can cause 0.5+E/2 ulp error where  in specialcase()
53 	 E is the worst-case ulp error outside the subnormal range.  So this  in specialcase()
54 	 is only useful if the goal is better than 1 ulp worst-case error.  */  in specialcase()
56       lo = scale - y + scale * tmp;  in specialcase()
58       lo = 1.0 - hi + y + lo;  in specialcase()
59       y = eval_as_double (hi + lo) - 1.0;  in specialcase()
60       /* Avoid -0.0 with downward rounding.  */  in specialcase()
64       force_eval_double (opt_barrier_double (0x1p-1022) * 0x1p-1022);  in specialcase()
66   y = 0x1p-1022 * y;  in specialcase()
85   abstop = top12 (x) & 0x7ff;  in exp2()
86   if (unlikely (abstop - top12 (0x1p-54) >= top12 (512.0) - top12 (0x1p-54)))  in exp2()
88       if (abstop - top12 (0x1p-54) >= 0x80000000)  in exp2()
90 	/* Note: 0 is common input.  */  in exp2()
94 	  if (asuint64 (x) == asuint64 (-INFINITY))  in exp2()
99 	    return __math_oflow (0);  in exp2()
100 	  else if (asuint64 (x) >= asuint64 (-1075.0))  in exp2()
101 	    return __math_uflow (0);  in exp2()
105 	abstop = 0;  in exp2()
108   /* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)].  */  in exp2()
109   /* x = k/N + r, with int k and r in [-1/2N, 1/2N].  */  in exp2()
112   kd -= Shift; /* k/N for int k.  */  in exp2()
113   r = x - kd;  in exp2()
116   top = ki << (52 - EXP_TABLE_BITS);  in exp2()
118   /* This is only a valid scale when -1023*N < k < 1024*N.  */  in exp2()
120   /* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1).  */  in exp2()
123   /* Without fma the worst case error is 0.5/N ulp larger.  */  in exp2()
124   /* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp.  */  in exp2()
132   if (unlikely (abstop == 0))  in exp2()
135   /* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there  in exp2()
147 TEST_SIG (S, D, 1, exp2, -9.9, 9.9)
150 TEST_INTERVAL (exp2, 0, 0xffff000000000000, 10000)
151 TEST_SYM_INTERVAL (exp2, 0x1p-6, 0x1p6, 40000)