Average Error: 43.6 → 0.8
Time: 51.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(\frac{1}{3} \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(\frac{1}{3} \cdot im\right)\right)
double f(double re, double im) {
        double r10658060 = 0.5;
        double r10658061 = re;
        double r10658062 = sin(r10658061);
        double r10658063 = r10658060 * r10658062;
        double r10658064 = im;
        double r10658065 = -r10658064;
        double r10658066 = exp(r10658065);
        double r10658067 = exp(r10658064);
        double r10658068 = r10658066 - r10658067;
        double r10658069 = r10658063 * r10658068;
        return r10658069;
}
↓
double f(double re, double im) {
        double r10658070 = 0.5;
        double r10658071 = re;
        double r10658072 = sin(r10658071);
        double r10658073 = r10658070 * r10658072;
        double r10658074 = -0.016666666666666666;
        double r10658075 = im;
        double r10658076 = 5.0;
        double r10658077 = pow(r10658075, r10658076);
        double r10658078 = r10658074 * r10658077;
        double r10658079 = r10658075 + r10658075;
        double r10658080 = r10658078 - r10658079;
        double r10658081 = r10658075 * r10658075;
        double r10658082 = 0.3333333333333333;
        double r10658083 = r10658082 * r10658075;
        double r10658084 = r10658081 * r10658083;
        double r10658085 = r10658080 - r10658084;
        double r10658086 = r10658073 * r10658085;
        return r10658086;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.6
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.6

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

  2. Taylor expanded around 0 0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]

  3. Simplified0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(\left({im}^{5} \cdot \frac{-1}{60} - \left(im + im\right)\right) - \left(im \cdot \frac{1}{3}\right) \cdot \left(im \cdot im\right)\right)}\]

  4. Final simplification0.8

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\left(\frac{-1}{60} \cdot {im}^{5} - \left(im + im\right)\right) - \left(im \cdot im\right) \cdot \left(\frac{1}{3} \cdot im\right)\right)\]

Reproduce

herbie shell --seed 2019200 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

  (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))