Average Error: 0.0 → 0.0
Time: 18.2s
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 r1648678 = 0.5;
        double r1648679 = re;
        double r1648680 = cos(r1648679);
        double r1648681 = r1648678 * r1648680;
        double r1648682 = im;
        double r1648683 = -r1648682;
        double r1648684 = exp(r1648683);
        double r1648685 = exp(r1648682);
        double r1648686 = r1648684 + r1648685;
        double r1648687 = r1648681 * r1648686;
        return r1648687;
}
↓
double f(double re, double im) {
        double r1648688 = re;
        double r1648689 = cos(r1648688);
        double r1648690 = im;
        double r1648691 = exp(r1648690);
        double r1648692 = 0.5;
        double r1648693 = r1648692 / r1648691;
        double r1648694 = fma(r1648691, r1648692, r1648693);
        double r1648695 = r1648689 * r1648694;
        return r1648695;
}

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 2019164 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))