Average Error: 0.0 → 0.2
Time: 4.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\sqrt{\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right)} \cdot \left(\sqrt{\mathsf{fma}\left(0.5, e^{im}, \log \left(e^{\frac{0.5}{e^{im}}}\right)\right)} \cdot \cos re\right)\]
\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}, \log \left(e^{\frac{0.5}{e^{im}}}\right)\right)} \cdot \cos re\right)
double f(double re, double im) {
        double r58558 = 0.5;
        double r58559 = re;
        double r58560 = cos(r58559);
        double r58561 = r58558 * r58560;
        double r58562 = im;
        double r58563 = -r58562;
        double r58564 = exp(r58563);
        double r58565 = exp(r58562);
        double r58566 = r58564 + r58565;
        double r58567 = r58561 * r58566;
        return r58567;
}
↓
double f(double re, double im) {
        double r58568 = 0.5;
        double r58569 = im;
        double r58570 = exp(r58569);
        double r58571 = r58568 / r58570;
        double r58572 = fma(r58568, r58570, r58571);
        double r58573 = sqrt(r58572);
        double r58574 = exp(r58571);
        double r58575 = log(r58574);
        double r58576 = fma(r58568, r58570, r58575);
        double r58577 = sqrt(r58576);
        double r58578 = re;
        double r58579 = cos(r58578);
        double r58580 = r58577 * r58579;
        double r58581 = r58573 * r58580;
        return r58581;
}

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. Using strategy rm

  7. Applied add-log-exp0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))