Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \cos re\right)
double f(double re, double im) {
        double r92806 = 0.5;
        double r92807 = re;
        double r92808 = cos(r92807);
        double r92809 = r92806 * r92808;
        double r92810 = im;
        double r92811 = -r92810;
        double r92812 = exp(r92811);
        double r92813 = exp(r92810);
        double r92814 = r92812 + r92813;
        double r92815 = r92809 * r92814;
        return r92815;
}
↓
double f(double re, double im) {
        double r92816 = 0.5;
        double r92817 = im;
        double r92818 = exp(r92817);
        double r92819 = r92816 / r92818;
        double r92820 = fma(r92816, r92818, r92819);
        double r92821 = sqrt(r92820);
        double r92822 = re;
        double r92823 = cos(r92822);
        double r92824 = r92821 * r92823;
        double r92825 = r92821 * r92824;
        return r92825;
}

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

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-*l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))