Average Error: 0.0 → 0.0
Time: 23.5s
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 r1211335 = 0.5;
        double r1211336 = re;
        double r1211337 = cos(r1211336);
        double r1211338 = r1211335 * r1211337;
        double r1211339 = im;
        double r1211340 = -r1211339;
        double r1211341 = exp(r1211340);
        double r1211342 = exp(r1211339);
        double r1211343 = r1211341 + r1211342;
        double r1211344 = r1211338 * r1211343;
        return r1211344;
}
↓
double f(double re, double im) {
        double r1211345 = re;
        double r1211346 = cos(r1211345);
        double r1211347 = im;
        double r1211348 = exp(r1211347);
        double r1211349 = 0.5;
        double r1211350 = r1211349 / r1211348;
        double r1211351 = fma(r1211348, r1211349, r1211350);
        double r1211352 = r1211346 * r1211351;
        return r1211352;
}

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 2019165 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))