Average Error: 0.0 → 0.0
Time: 17.9s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\sqrt[3]{e^{-im}} \cdot \sqrt[3]{e^{-im}}, \sqrt[3]{e^{-im}}, e^{im}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\sqrt[3]{e^{-im}} \cdot \sqrt[3]{e^{-im}}, \sqrt[3]{e^{-im}}, e^{im}\right)
double f(double re, double im) {
        double r42505 = 0.5;
        double r42506 = re;
        double r42507 = cos(r42506);
        double r42508 = r42505 * r42507;
        double r42509 = im;
        double r42510 = -r42509;
        double r42511 = exp(r42510);
        double r42512 = exp(r42509);
        double r42513 = r42511 + r42512;
        double r42514 = r42508 * r42513;
        return r42514;
}
↓
double f(double re, double im) {
        double r42515 = 0.5;
        double r42516 = re;
        double r42517 = cos(r42516);
        double r42518 = r42515 * r42517;
        double r42519 = im;
        double r42520 = -r42519;
        double r42521 = exp(r42520);
        double r42522 = cbrt(r42521);
        double r42523 = r42522 * r42522;
        double r42524 = exp(r42519);
        double r42525 = fma(r42523, r42522, r42524);
        double r42526 = r42518 * r42525;
        return r42526;
}

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 add-cube-cbrt0.0

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(\color{blue}{\left(\sqrt[3]{e^{-im}} \cdot \sqrt[3]{e^{-im}}\right) \cdot \sqrt[3]{e^{-im}}} + e^{im}\right)\]

  4. Applied fma-def0.0

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{-im}} \cdot \sqrt[3]{e^{-im}}, \sqrt[3]{e^{-im}}, e^{im}\right)}\]

  5. Final simplification0.0

    \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \mathsf{fma}\left(\sqrt[3]{e^{-im}} \cdot \sqrt[3]{e^{-im}}, \sqrt[3]{e^{-im}}, e^{im}\right)\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))