Average Error: 0.0 → 0.0
Time: 18.3s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r30466 = 0.5;
        double r30467 = re;
        double r30468 = cos(r30467);
        double r30469 = r30466 * r30468;
        double r30470 = im;
        double r30471 = -r30470;
        double r30472 = exp(r30471);
        double r30473 = exp(r30470);
        double r30474 = r30472 + r30473;
        double r30475 = r30469 * r30474;
        return r30475;
}
↓
double f(double re, double im) {
        double r30476 = re;
        double r30477 = cos(r30476);
        double r30478 = im;
        double r30479 = exp(r30478);
        double r30480 = r30477 / r30479;
        double r30481 = fma(r30477, r30479, r30480);
        double r30482 = 0.5;
        double r30483 = r30481 * r30482;
        return r30483;
}

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 distribute-lft-in0.0

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

  4. Simplified0.0

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

  5. Taylor expanded around inf 0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))