Average Error: 0.0 → 0.0
Time: 1.4m
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left(e^{im}\right), \left(\cos re \cdot 0.5\right), \left(\frac{\cos re \cdot 0.5}{e^{im}}\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\left(e^{im}\right), \left(\cos re \cdot 0.5\right), \left(\frac{\cos re \cdot 0.5}{e^{im}}\right)\right)
double f(double re, double im) {
        double r10797847 = 0.5;
        double r10797848 = re;
        double r10797849 = cos(r10797848);
        double r10797850 = r10797847 * r10797849;
        double r10797851 = im;
        double r10797852 = -r10797851;
        double r10797853 = exp(r10797852);
        double r10797854 = exp(r10797851);
        double r10797855 = r10797853 + r10797854;
        double r10797856 = r10797850 * r10797855;
        return r10797856;
}
↓
double f(double re, double im) {
        double r10797857 = im;
        double r10797858 = exp(r10797857);
        double r10797859 = re;
        double r10797860 = cos(r10797859);
        double r10797861 = 0.5;
        double r10797862 = r10797860 * r10797861;
        double r10797863 = r10797862 / r10797858;
        double r10797864 = fma(r10797858, r10797862, r10797863);
        return r10797864;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))