math.cos on complex, real part

Average Error: 0.0 → 0.0

Time: 16.1s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r1081765 = 0.5;
        double r1081766 = re;
        double r1081767 = cos(r1081766);
        double r1081768 = r1081765 * r1081767;
        double r1081769 = im;
        double r1081770 = -r1081769;
        double r1081771 = exp(r1081770);
        double r1081772 = exp(r1081769);
        double r1081773 = r1081771 + r1081772;
        double r1081774 = r1081768 * r1081773;
        return r1081774;
}

↓

double f(double re, double im) {
        double r1081775 = 0.5;
        double r1081776 = re;
        double r1081777 = cos(r1081776);
        double r1081778 = im;
        double r1081779 = exp(r1081778);
        double r1081780 = r1081777 / r1081779;
        double r1081781 = fma(r1081777, r1081779, r1081780);
        double r1081782 = r1081775 * r1081781;
        return r1081782;
}

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

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))