Average Error: 0.0 → 0.0
Time: 13.6s
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 r1517025 = 0.5;
        double r1517026 = re;
        double r1517027 = cos(r1517026);
        double r1517028 = r1517025 * r1517027;
        double r1517029 = im;
        double r1517030 = -r1517029;
        double r1517031 = exp(r1517030);
        double r1517032 = exp(r1517029);
        double r1517033 = r1517031 + r1517032;
        double r1517034 = r1517028 * r1517033;
        return r1517034;
}
↓
double f(double re, double im) {
        double r1517035 = re;
        double r1517036 = cos(r1517035);
        double r1517037 = im;
        double r1517038 = exp(r1517037);
        double r1517039 = 0.5;
        double r1517040 = r1517039 / r1517038;
        double r1517041 = fma(r1517038, r1517039, r1517040);
        double r1517042 = r1517036 * r1517041;
        return r1517042;
}

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 2019169 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))