Average Error: 0.0 → 0.0
Time: 15.5s
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 r1260527 = 0.5;
        double r1260528 = re;
        double r1260529 = cos(r1260528);
        double r1260530 = r1260527 * r1260529;
        double r1260531 = im;
        double r1260532 = -r1260531;
        double r1260533 = exp(r1260532);
        double r1260534 = exp(r1260531);
        double r1260535 = r1260533 + r1260534;
        double r1260536 = r1260530 * r1260535;
        return r1260536;
}
↓
double f(double re, double im) {
        double r1260537 = re;
        double r1260538 = cos(r1260537);
        double r1260539 = im;
        double r1260540 = exp(r1260539);
        double r1260541 = 0.5;
        double r1260542 = r1260541 / r1260540;
        double r1260543 = fma(r1260540, r1260541, r1260542);
        double r1260544 = r1260538 * r1260543;
        return r1260544;
}

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 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))