Average Error: 0.0 → 0.0
Time: 5.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, 0.5 \cdot \cos re, \frac{0.5 \cdot \cos re}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, 0.5 \cdot \cos re, \frac{0.5 \cdot \cos re}{e^{im}}\right)
double f(double re, double im) {
        double r61419 = 0.5;
        double r61420 = re;
        double r61421 = cos(r61420);
        double r61422 = r61419 * r61421;
        double r61423 = im;
        double r61424 = -r61423;
        double r61425 = exp(r61424);
        double r61426 = exp(r61423);
        double r61427 = r61425 + r61426;
        double r61428 = r61422 * r61427;
        return r61428;
}
↓
double f(double re, double im) {
        double r61429 = im;
        double r61430 = exp(r61429);
        double r61431 = 0.5;
        double r61432 = re;
        double r61433 = cos(r61432);
        double r61434 = r61431 * r61433;
        double r61435 = r61434 / r61430;
        double r61436 = fma(r61430, r61434, r61435);
        return r61436;
}

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 *-un-lft-identity0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{\color{blue}{1 \cdot e^{im}}}\right) \cdot \cos re\]

  5. Applied add-sqr-sqrt0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \frac{\color{blue}{\sqrt{0.5} \cdot \sqrt{0.5}}}{1 \cdot e^{im}}\right) \cdot \cos re\]

  6. Applied times-frac0.0

    \[\leadsto \mathsf{fma}\left(0.5, e^{im}, \color{blue}{\frac{\sqrt{0.5}}{1} \cdot \frac{\sqrt{0.5}}{e^{im}}}\right) \cdot \cos re\]

  7. Simplified0.0

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

  9. Applied *-un-lft-identity0.0

    \[\leadsto \color{blue}{\left(1 \cdot \mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right)\right)} \cdot \cos re\]

  10. Applied associate-*l*0.0

    \[\leadsto \color{blue}{1 \cdot \left(\mathsf{fma}\left(0.5, e^{im}, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re\right)}\]

  11. Simplified0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(e^{im}, 0.5 \cdot \cos re, \frac{0.5 \cdot \cos re}{e^{im}}\right)}\]

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))