Average Error: 0.0 → 0.0
Time: 5.6s
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 r66308 = 0.5;
        double r66309 = re;
        double r66310 = cos(r66309);
        double r66311 = r66308 * r66310;
        double r66312 = im;
        double r66313 = -r66312;
        double r66314 = exp(r66313);
        double r66315 = exp(r66312);
        double r66316 = r66314 + r66315;
        double r66317 = r66311 * r66316;
        return r66317;
}
↓
double f(double re, double im) {
        double r66318 = im;
        double r66319 = exp(r66318);
        double r66320 = 0.5;
        double r66321 = re;
        double r66322 = cos(r66321);
        double r66323 = r66320 * r66322;
        double r66324 = r66323 / r66319;
        double r66325 = fma(r66319, r66323, r66324);
        return r66325;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{\cos re \cdot \left(\frac{{\left(\sqrt{0.5}\right)}^{2}}{e^{im}} + 0.5 \cdot e^{im}\right)}\]

  9. Simplified0.0

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

  10. 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))))