Average Error: 0.0 → 0.0
Time: 25.6s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[(\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
(\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*
double f(double re, double im) {
        double r1930645 = 0.5;
        double r1930646 = re;
        double r1930647 = cos(r1930646);
        double r1930648 = r1930645 * r1930647;
        double r1930649 = im;
        double r1930650 = -r1930649;
        double r1930651 = exp(r1930650);
        double r1930652 = exp(r1930649);
        double r1930653 = r1930651 + r1930652;
        double r1930654 = r1930648 * r1930653;
        return r1930654;
}
↓
double f(double re, double im) {
        double r1930655 = im;
        double r1930656 = exp(r1930655);
        double r1930657 = re;
        double r1930658 = cos(r1930657);
        double r1930659 = 0.5;
        double r1930660 = r1930658 * r1930659;
        double r1930661 = r1930660 / r1930656;
        double r1930662 = fma(r1930656, r1930660, r1930661);
        return r1930662;
}

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

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

  3. Final simplification0.0

    \[\leadsto (\left(e^{im}\right) \cdot \left(\cos re \cdot 0.5\right) + \left(\frac{\cos re \cdot 0.5}{e^{im}}\right))_*\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (re im)
  :name "math.cos on complex, real part"
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))