Average Error: 0.0 → 0.0
Time: 14.5s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\left(e^{im}\right), \left(\cos re \cdot 0.5\right), \left(\frac{\cos re \cdot 0.5}{e^{im}}\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\left(e^{im}\right), \left(\cos re \cdot 0.5\right), \left(\frac{\cos re \cdot 0.5}{e^{im}}\right)\right)
double f(double re, double im) {
        double r1275547 = 0.5;
        double r1275548 = re;
        double r1275549 = cos(r1275548);
        double r1275550 = r1275547 * r1275549;
        double r1275551 = im;
        double r1275552 = -r1275551;
        double r1275553 = exp(r1275552);
        double r1275554 = exp(r1275551);
        double r1275555 = r1275553 + r1275554;
        double r1275556 = r1275550 * r1275555;
        return r1275556;
}
↓
double f(double re, double im) {
        double r1275557 = im;
        double r1275558 = exp(r1275557);
        double r1275559 = re;
        double r1275560 = cos(r1275559);
        double r1275561 = 0.5;
        double r1275562 = r1275560 * r1275561;
        double r1275563 = r1275562 / r1275558;
        double r1275564 = fma(r1275558, r1275562, r1275563);
        return r1275564;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))