math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 16.3s

Precision: 64

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]

↓

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

C

double f(double re, double im) {
        double r1071638 = 0.5;
        double r1071639 = re;
        double r1071640 = cos(r1071639);
        double r1071641 = r1071638 * r1071640;
        double r1071642 = im;
        double r1071643 = -r1071642;
        double r1071644 = exp(r1071643);
        double r1071645 = exp(r1071642);
        double r1071646 = r1071644 + r1071645;
        double r1071647 = r1071641 * r1071646;
        return r1071647;
}

↓

double f(double re, double im) {
        double r1071648 = 0.5;
        double r1071649 = im;
        double r1071650 = exp(r1071649);
        double r1071651 = re;
        double r1071652 = cos(r1071651);
        double r1071653 = r1071652 / r1071650;
        double r1071654 = fma(r1071650, r1071652, r1071653);
        double r1071655 = r1071648 * r1071654;
        return r1071655;
}

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. Using strategy rm

3. Applied associate-*l* 0.0

   \[\leadsto \color{blue}{0.5 \cdot \left(\cos re \cdot \left(e^{-im} + e^{im}\right)\right)}\]

4. Simplified 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))