Average Error: 0.0 → 0.0
Time: 12.9s
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 r1471330 = 0.5;
        double r1471331 = re;
        double r1471332 = cos(r1471331);
        double r1471333 = r1471330 * r1471332;
        double r1471334 = im;
        double r1471335 = -r1471334;
        double r1471336 = exp(r1471335);
        double r1471337 = exp(r1471334);
        double r1471338 = r1471336 + r1471337;
        double r1471339 = r1471333 * r1471338;
        return r1471339;
}
↓
double f(double re, double im) {
        double r1471340 = im;
        double r1471341 = exp(r1471340);
        double r1471342 = re;
        double r1471343 = cos(r1471342);
        double r1471344 = 0.5;
        double r1471345 = r1471343 * r1471344;
        double r1471346 = r1471345 / r1471341;
        double r1471347 = fma(r1471341, r1471345, r1471346);
        return r1471347;
}

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 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))