Average Error: 0.0 → 0.0
Time: 13.0s
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 r723561 = 0.5;
        double r723562 = re;
        double r723563 = cos(r723562);
        double r723564 = r723561 * r723563;
        double r723565 = im;
        double r723566 = -r723565;
        double r723567 = exp(r723566);
        double r723568 = exp(r723565);
        double r723569 = r723567 + r723568;
        double r723570 = r723564 * r723569;
        return r723570;
}
↓
double f(double re, double im) {
        double r723571 = re;
        double r723572 = cos(r723571);
        double r723573 = im;
        double r723574 = exp(r723573);
        double r723575 = 0.5;
        double r723576 = r723575 / r723574;
        double r723577 = fma(r723574, r723575, r723576);
        double r723578 = r723572 * r723577;
        return r723578;
}

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 2019153 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))