Average Error: 0.0 → 0.0
Time: 20.3s
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 r2027647 = 0.5;
        double r2027648 = re;
        double r2027649 = cos(r2027648);
        double r2027650 = r2027647 * r2027649;
        double r2027651 = im;
        double r2027652 = -r2027651;
        double r2027653 = exp(r2027652);
        double r2027654 = exp(r2027651);
        double r2027655 = r2027653 + r2027654;
        double r2027656 = r2027650 * r2027655;
        return r2027656;
}
↓
double f(double re, double im) {
        double r2027657 = re;
        double r2027658 = cos(r2027657);
        double r2027659 = im;
        double r2027660 = exp(r2027659);
        double r2027661 = 0.5;
        double r2027662 = r2027661 / r2027660;
        double r2027663 = fma(r2027660, r2027661, r2027662);
        double r2027664 = r2027658 * r2027663;
        return r2027664;
}

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 2019139 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))