Average Error: 0.0 → 0.0
Time: 20.2s
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 r36974 = 0.5;
        double r36975 = re;
        double r36976 = cos(r36975);
        double r36977 = r36974 * r36976;
        double r36978 = im;
        double r36979 = -r36978;
        double r36980 = exp(r36979);
        double r36981 = exp(r36978);
        double r36982 = r36980 + r36981;
        double r36983 = r36977 * r36982;
        return r36983;
}
↓
double f(double re, double im) {
        double r36984 = 0.5;
        double r36985 = im;
        double r36986 = exp(r36985);
        double r36987 = re;
        double r36988 = cos(r36987);
        double r36989 = r36988 / r36986;
        double r36990 = fma(r36986, r36988, r36989);
        double r36991 = r36984 * r36990;
        return r36991;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.0

    \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot e^{-im} + \left(0.5 \cdot \cos re\right) \cdot e^{im}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{0.5 \cdot \frac{\cos re}{e^{im}}} + \left(0.5 \cdot \cos re\right) \cdot e^{im}\]

  5. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{0.5 \cdot \left(e^{im} \cdot \cos re\right) + 0.5 \cdot \frac{\cos re}{e^{im}}}\]

  6. Simplified0.0

    \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)}\]

  7. Final simplification0.0

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))