Average Error: 0.0 → 0.7
Time: 8.7s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)\]
↓
\[\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\frac{1}{12} \cdot {im}^{4} + \left({im}^{2} + 2\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)
↓
\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\frac{1}{12} \cdot {im}^{4} + \left({im}^{2} + 2\right)\right)
double f(double re, double im) {
        double r63601 = 0.5;
        double r63602 = re;
        double r63603 = cos(r63602);
        double r63604 = r63601 * r63603;
        double r63605 = im;
        double r63606 = -r63605;
        double r63607 = exp(r63606);
        double r63608 = exp(r63605);
        double r63609 = r63607 + r63608;
        double r63610 = r63604 * r63609;
        return r63610;
}
↓
double f(double re, double im) {
        double r63611 = 1.0;
        double r63612 = 2.0;
        double r63613 = r63611 / r63612;
        double r63614 = re;
        double r63615 = cos(r63614);
        double r63616 = r63613 * r63615;
        double r63617 = 0.08333333333333333;
        double r63618 = im;
        double r63619 = 4.0;
        double r63620 = pow(r63618, r63619);
        double r63621 = r63617 * r63620;
        double r63622 = 2.0;
        double r63623 = pow(r63618, r63622);
        double r63624 = r63623 + r63622;
        double r63625 = r63621 + r63624;
        double r63626 = r63616 * r63625;
        return r63626;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)}\]

  3. Taylor expanded around 0 0.7

    \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \color{blue}{\left(\frac{1}{12} \cdot {im}^{4} + \left({im}^{2} + 2\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto \left(\frac{1}{2} \cdot \cos re\right) \cdot \left(\frac{1}{12} \cdot {im}^{4} + \left({im}^{2} + 2\right)\right)\]

Reproduce

herbie shell --seed 2019304 
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))