Average Error: 0.0 → 0.0
Time: 44.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)
double f(double re, double im) {
        double r1131439 = 0.5;
        double r1131440 = re;
        double r1131441 = cos(r1131440);
        double r1131442 = r1131439 * r1131441;
        double r1131443 = im;
        double r1131444 = -r1131443;
        double r1131445 = exp(r1131444);
        double r1131446 = exp(r1131443);
        double r1131447 = r1131445 + r1131446;
        double r1131448 = r1131442 * r1131447;
        return r1131448;
}
↓
double f(double re, double im) {
        double r1131449 = 0.5;
        double r1131450 = im;
        double r1131451 = exp(r1131450);
        double r1131452 = re;
        double r1131453 = cos(r1131452);
        double r1131454 = r1131453 / r1131451;
        double r1131455 = fma(r1131451, r1131453, r1131454);
        double r1131456 = r1131449 * r1131455;
        return r1131456;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))