1use super::super::{CastFrom, CastInto, Float, IntTy, MinInt};
2
3/// Scale the exponent.
4///
5/// From N3220:
6///
7/// > The scalbn and scalbln functions compute `x * b^n`, where `b = FLT_RADIX` if the return type
8/// > of the function is a standard floating type, or `b = 10` if the return type of the function
9/// > is a decimal floating type. A range error occurs for some finite x, depending on n.
10/// >
11/// > [...]
12/// >
13/// > * `scalbn(±0, n)` returns `±0`.
14/// > * `scalbn(x, 0)` returns `x`.
15/// > * `scalbn(±∞, n)` returns `±∞`.
16/// >
17/// > If the calculation does not overflow or underflow, the returned value is exact and
18/// > independent of the current rounding direction mode.
19#[inline]
20pub fn scalbn<F: Float>(mut x: F, mut n: i32) -> F
21where
22 u32: CastInto<F::Int>,
23 F::Int: CastFrom<i32>,
24 F::Int: CastFrom<u32>,
25{
26 let zero = IntTy::<F>::ZERO;
27
28 // Bits including the implicit bit
29 let sig_total_bits = F::SIG_BITS + 1;
30
31 // Maximum and minimum values when biased
32 let exp_max = F::EXP_MAX;
33 let exp_min = F::EXP_MIN;
34
35 // 2 ^ Emax, maximum positive with null significand (0x1p1023 for f64)
36 let f_exp_max = F::from_parts(false, F::EXP_BIAS << 1, zero);
37
38 // 2 ^ Emin, minimum positive normal with null significand (0x1p-1022 for f64)
39 let f_exp_min = F::from_parts(false, 1, zero);
40
41 // 2 ^ sig_total_bits, moltiplier to normalize subnormals (0x1p53 for f64)
42 let f_pow_subnorm = F::from_parts(false, sig_total_bits + F::EXP_BIAS, zero);
43
44 /*
45 * The goal is to multiply `x` by a scale factor that applies `n`. However, there are cases
46 * where `2^n` is not representable by `F` but the result should be, e.g. `x = 2^Emin` with
47 * `n = -EMin + 2` (one out of range of 2^Emax). To get around this, reduce the magnitude of
48 * the final scale operation by prescaling by the max/min power representable by `F`.
49 */
50
51 if n > exp_max {
52 // Worse case positive `n`: `x` is the minimum subnormal value, the result is `F::MAX`.
53 // This can be reached by three scaling multiplications (two here and one final).
54 debug_assert!(-exp_min + F::SIG_BITS as i32 + exp_max <= exp_max * 3);
55
56 x *= f_exp_max;
57 n -= exp_max;
58 if n > exp_max {
59 x *= f_exp_max;
60 n -= exp_max;
61 if n > exp_max {
62 n = exp_max;
63 }
64 }
65 } else if n < exp_min {
66 // When scaling toward 0, the prescaling is limited to a value that does not allow `x` to
67 // go subnormal. This avoids double rounding.
68 if F::BITS > 16 {
69 // `mul` s.t. `!(x * mul).is_subnormal() ∀ x`
70 let mul = f_exp_min * f_pow_subnorm;
71 let add = -exp_min - sig_total_bits as i32;
72
73 // Worse case negative `n`: `x` is the maximum positive value, the result is `F::MIN`.
74 // This must be reachable by three scaling multiplications (two here and one final).
75 debug_assert!(-exp_min + F::SIG_BITS as i32 + exp_max <= add * 2 + -exp_min);
76
77 x *= mul;
78 n += add;
79
80 if n < exp_min {
81 x *= mul;
82 n += add;
83
84 if n < exp_min {
85 n = exp_min;
86 }
87 }
88 } else {
89 // `f16` is unique compared to other float types in that the difference between the
90 // minimum exponent and the significand bits (`add = -exp_min - sig_total_bits`) is
91 // small, only three. The above method depend on decrementing `n` by `add` two times;
92 // for other float types this works out because `add` is a substantial fraction of
93 // the exponent range. For `f16`, however, 3 is relatively small compared to the
94 // exponent range (which is 39), so that requires ~10 prescale rounds rather than two.
95 //
96 // Work aroudn this by using a different algorithm that calculates the prescale
97 // dynamically based on the maximum possible value. This adds more operations per round
98 // since it needs to construct the scale, but works better in the general case.
99 let add = -(n + sig_total_bits as i32).clamp(exp_min, sig_total_bits as i32);
100 let mul = F::from_parts(false, (F::EXP_BIAS as i32 - add) as u32, zero);
101
102 x *= mul;
103 n += add;
104
105 if n < exp_min {
106 let add = -(n + sig_total_bits as i32).clamp(exp_min, sig_total_bits as i32);
107 let mul = F::from_parts(false, (F::EXP_BIAS as i32 - add) as u32, zero);
108
109 x *= mul;
110 n += add;
111
112 if n < exp_min {
113 n = exp_min;
114 }
115 }
116 }
117 }
118
119 let scale = F::from_parts(false, (F::EXP_BIAS as i32 + n) as u32, zero);
120 x * scale
121}
122