Average Error: 0.0 → 0.0
Time: 15.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\cos re \cdot \mathsf{fma}\left(e^{im}, 0.5, \frac{0.5}{e^{im}}\right)
double f(double re, double im) {
        double r1260529 = 0.5;
        double r1260530 = re;
        double r1260531 = cos(r1260530);
        double r1260532 = r1260529 * r1260531;
        double r1260533 = im;
        double r1260534 = -r1260533;
        double r1260535 = exp(r1260534);
        double r1260536 = exp(r1260533);
        double r1260537 = r1260535 + r1260536;
        double r1260538 = r1260532 * r1260537;
        return r1260538;
}
↓
double f(double re, double im) {
        double r1260539 = re;
        double r1260540 = cos(r1260539);
        double r1260541 = im;
        double r1260542 = exp(r1260541);
        double r1260543 = 0.5;
        double r1260544 = r1260543 / r1260542;
        double r1260545 = fma(r1260542, r1260543, r1260544);
        double r1260546 = r1260540 * r1260545;
        return r1260546;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))