Average Error: 0.0 → 0.0
Time: 13.2s
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 r1563264 = 0.5;
        double r1563265 = re;
        double r1563266 = cos(r1563265);
        double r1563267 = r1563264 * r1563266;
        double r1563268 = im;
        double r1563269 = -r1563268;
        double r1563270 = exp(r1563269);
        double r1563271 = exp(r1563268);
        double r1563272 = r1563270 + r1563271;
        double r1563273 = r1563267 * r1563272;
        return r1563273;
}
↓
double f(double re, double im) {
        double r1563274 = re;
        double r1563275 = cos(r1563274);
        double r1563276 = im;
        double r1563277 = exp(r1563276);
        double r1563278 = 0.5;
        double r1563279 = r1563278 / r1563277;
        double r1563280 = fma(r1563277, r1563278, r1563279);
        double r1563281 = r1563275 * r1563280;
        return r1563281;
}

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 2019168 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))