Average Error: 0.0 → 0.0
Time: 13.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\mathsf{fma}\left(\cos re, e^{im}, \frac{\cos re}{e^{im}}\right) \cdot 0.5
double f(double re, double im) {
        double r1602338 = 0.5;
        double r1602339 = re;
        double r1602340 = cos(r1602339);
        double r1602341 = r1602338 * r1602340;
        double r1602342 = im;
        double r1602343 = -r1602342;
        double r1602344 = exp(r1602343);
        double r1602345 = exp(r1602342);
        double r1602346 = r1602344 + r1602345;
        double r1602347 = r1602341 * r1602346;
        return r1602347;
}
↓
double f(double re, double im) {
        double r1602348 = re;
        double r1602349 = cos(r1602348);
        double r1602350 = im;
        double r1602351 = exp(r1602350);
        double r1602352 = r1602349 / r1602351;
        double r1602353 = fma(r1602349, r1602351, r1602352);
        double r1602354 = 0.5;
        double r1602355 = r1602353 * r1602354;
        return r1602355;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))