Average Error: 0.0 → 0.0
Time: 25.7s
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 r2135420 = 0.5;
        double r2135421 = re;
        double r2135422 = cos(r2135421);
        double r2135423 = r2135420 * r2135422;
        double r2135424 = im;
        double r2135425 = -r2135424;
        double r2135426 = exp(r2135425);
        double r2135427 = exp(r2135424);
        double r2135428 = r2135426 + r2135427;
        double r2135429 = r2135423 * r2135428;
        return r2135429;
}
↓
double f(double re, double im) {
        double r2135430 = re;
        double r2135431 = cos(r2135430);
        double r2135432 = im;
        double r2135433 = exp(r2135432);
        double r2135434 = 0.5;
        double r2135435 = r2135434 / r2135433;
        double r2135436 = fma(r2135433, r2135434, r2135435);
        double r2135437 = r2135431 * r2135436;
        return r2135437;
}

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 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))