Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(0.5, e^{im}, \frac{0.5}{e^{im}}\right) \cdot \cos re
double f(double re, double im) {
        double r49441 = 0.5;
        double r49442 = re;
        double r49443 = cos(r49442);
        double r49444 = r49441 * r49443;
        double r49445 = im;
        double r49446 = -r49445;
        double r49447 = exp(r49446);
        double r49448 = exp(r49445);
        double r49449 = r49447 + r49448;
        double r49450 = r49444 * r49449;
        return r49450;
}
↓
double f(double re, double im) {
        double r49451 = 0.5;
        double r49452 = im;
        double r49453 = exp(r49452);
        double r49454 = r49451 / r49453;
        double r49455 = fma(r49451, r49453, r49454);
        double r49456 = re;
        double r49457 = cos(r49456);
        double r49458 = r49455 * r49457;
        return r49458;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))