Average Error: 0.0 → 0.0
Time: 13.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, 0.5, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, 0.5, \sqrt{0.5} \cdot \frac{\sqrt{0.5}}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r872494 = 0.5;
        double r872495 = re;
        double r872496 = cos(r872495);
        double r872497 = r872494 * r872496;
        double r872498 = im;
        double r872499 = -r872498;
        double r872500 = exp(r872499);
        double r872501 = exp(r872498);
        double r872502 = r872500 + r872501;
        double r872503 = r872497 * r872502;
        return r872503;
}
↓
double f(double re, double im) {
        double r872504 = im;
        double r872505 = exp(r872504);
        double r872506 = 0.5;
        double r872507 = sqrt(r872506);
        double r872508 = r872507 / r872505;
        double r872509 = r872507 * r872508;
        double r872510 = fma(r872505, r872506, r872509);
        double r872511 = re;
        double r872512 = cos(r872511);
        double r872513 = r872510 * r872512;
        return r872513;
}

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. Using strategy rm

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

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))