Average Error: 0.0 → 0.0
Time: 17.8s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r2038451 = 0.5;
        double r2038452 = re;
        double r2038453 = cos(r2038452);
        double r2038454 = r2038451 * r2038453;
        double r2038455 = im;
        double r2038456 = -r2038455;
        double r2038457 = exp(r2038456);
        double r2038458 = exp(r2038455);
        double r2038459 = r2038457 + r2038458;
        double r2038460 = r2038454 * r2038459;
        return r2038460;
}
↓
double f(double re, double im) {
        double r2038461 = im;
        double r2038462 = exp(r2038461);
        double r2038463 = 0.5;
        double r2038464 = r2038463 / r2038462;
        double r2038465 = fma(r2038462, r2038463, r2038464);
        double r2038466 = re;
        double r2038467 = cos(r2038466);
        double r2038468 = r2038465 * r2038467;
        return r2038468;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))