Average Error: 0.0 → 0.0
Time: 3.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r44038 = 0.5;
        double r44039 = re;
        double r44040 = cos(r44039);
        double r44041 = r44038 * r44040;
        double r44042 = im;
        double r44043 = -r44042;
        double r44044 = exp(r44043);
        double r44045 = exp(r44042);
        double r44046 = r44044 + r44045;
        double r44047 = r44041 * r44046;
        return r44047;
}
↓
double f(double re, double im) {
        double r44048 = 0.5;
        double r44049 = im;
        double r44050 = exp(r44049);
        double r44051 = r44048 / r44050;
        double r44052 = fma(r44048, r44050, r44051);
        double r44053 = re;
        double r44054 = cos(r44053);
        double r44055 = r44052 * r44054;
        return r44055;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))