Average Error: 0.0 → 0.0
Time: 4.1s
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 r48285 = 0.5;
        double r48286 = re;
        double r48287 = cos(r48286);
        double r48288 = r48285 * r48287;
        double r48289 = im;
        double r48290 = -r48289;
        double r48291 = exp(r48290);
        double r48292 = exp(r48289);
        double r48293 = r48291 + r48292;
        double r48294 = r48288 * r48293;
        return r48294;
}
↓
double f(double re, double im) {
        double r48295 = 0.5;
        double r48296 = im;
        double r48297 = exp(r48296);
        double r48298 = r48295 / r48297;
        double r48299 = fma(r48295, r48297, r48298);
        double r48300 = re;
        double r48301 = cos(r48300);
        double r48302 = r48299 * r48301;
        return r48302;
}

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 2020049 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))