Average Error: 43.6 → 0.8
Time: 30.7s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\frac{-1}{3} \cdot \left(im \cdot \left(im \cdot im\right)\right) - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r11068042 = 0.5;
        double r11068043 = re;
        double r11068044 = sin(r11068043);
        double r11068045 = r11068042 * r11068044;
        double r11068046 = im;
        double r11068047 = -r11068046;
        double r11068048 = exp(r11068047);
        double r11068049 = exp(r11068046);
        double r11068050 = r11068048 - r11068049;
        double r11068051 = r11068045 * r11068050;
        return r11068051;
}
↓
double f(double re, double im) {
        double r11068052 = -0.3333333333333333;
        double r11068053 = im;
        double r11068054 = r11068053 * r11068053;
        double r11068055 = r11068053 * r11068054;
        double r11068056 = r11068052 * r11068055;
        double r11068057 = 5.0;
        double r11068058 = pow(r11068053, r11068057);
        double r11068059 = 0.016666666666666666;
        double r11068060 = r11068058 * r11068059;
        double r11068061 = r11068053 + r11068053;
        double r11068062 = r11068060 + r11068061;
        double r11068063 = r11068056 - r11068062;
        double r11068064 = 0.5;
        double r11068065 = re;
        double r11068066 = sin(r11068065);
        double r11068067 = r11068064 * r11068066;
        double r11068068 = r11068063 * r11068067;
        return r11068068;
}

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

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019174 
(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))))