Average Error: 0.0 → 0.0
Time: 10.4s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r745354 = 0.5;
        double r745355 = re;
        double r745356 = cos(r745355);
        double r745357 = r745354 * r745356;
        double r745358 = im;
        double r745359 = -r745358;
        double r745360 = exp(r745359);
        double r745361 = exp(r745358);
        double r745362 = r745360 + r745361;
        double r745363 = r745357 * r745362;
        return r745363;
}
↓
double f(double re, double im) {
        double r745364 = re;
        double r745365 = cos(r745364);
        double r745366 = im;
        double r745367 = exp(r745366);
        double r745368 = 0.5;
        double r745369 = r745368 / r745367;
        double r745370 = fma(r745367, r745368, r745369);
        double r745371 = r745365 * r745370;
        return r745371;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))