Average Error: 0.0 → 0.0
Time: 20.4s
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 r713268 = 0.5;
        double r713269 = re;
        double r713270 = cos(r713269);
        double r713271 = r713268 * r713270;
        double r713272 = im;
        double r713273 = -r713272;
        double r713274 = exp(r713273);
        double r713275 = exp(r713272);
        double r713276 = r713274 + r713275;
        double r713277 = r713271 * r713276;
        return r713277;
}
↓
double f(double re, double im) {
        double r713278 = 0.5;
        double r713279 = im;
        double r713280 = exp(r713279);
        double r713281 = re;
        double r713282 = cos(r713281);
        double r713283 = r713282 / r713280;
        double r713284 = fma(r713280, r713282, r713283);
        double r713285 = r713278 * r713284;
        return r713285;
}

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 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))