Average Error: 0.0 → 0.0
Time: 5.0s
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 r44552 = 0.5;
        double r44553 = re;
        double r44554 = cos(r44553);
        double r44555 = r44552 * r44554;
        double r44556 = im;
        double r44557 = -r44556;
        double r44558 = exp(r44557);
        double r44559 = exp(r44556);
        double r44560 = r44558 + r44559;
        double r44561 = r44555 * r44560;
        return r44561;
}
↓
double f(double re, double im) {
        double r44562 = 0.5;
        double r44563 = im;
        double r44564 = exp(r44563);
        double r44565 = r44562 / r44564;
        double r44566 = fma(r44562, r44564, r44565);
        double r44567 = re;
        double r44568 = cos(r44567);
        double r44569 = r44566 * r44568;
        return r44569;
}

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 2020062 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))