Average Error: 0.0 → 0.0
Time: 23.2s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
0.5 \cdot \mathsf{fma}\left(e^{im}, \cos re, \frac{\cos re}{e^{im}}\right)
double f(double re, double im) {
        double r1190406 = 0.5;
        double r1190407 = re;
        double r1190408 = cos(r1190407);
        double r1190409 = r1190406 * r1190408;
        double r1190410 = im;
        double r1190411 = -r1190410;
        double r1190412 = exp(r1190411);
        double r1190413 = exp(r1190410);
        double r1190414 = r1190412 + r1190413;
        double r1190415 = r1190409 * r1190414;
        return r1190415;
}
↓
double f(double re, double im) {
        double r1190416 = 0.5;
        double r1190417 = im;
        double r1190418 = exp(r1190417);
        double r1190419 = re;
        double r1190420 = cos(r1190419);
        double r1190421 = r1190420 / r1190418;
        double r1190422 = fma(r1190418, r1190420, r1190421);
        double r1190423 = r1190416 * r1190422;
        return r1190423;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))