libm/math/generic/
scalbn.rs

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1use crate::support::{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}
libm/math/generic/scalbn.rs

libm/math/generic/
scalbn.rs
