Average Error: 58.0 → 0.7
Time: 32.1s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)
↓
\left(\left(im \cdot \left(im \cdot im\right)\right) \cdot \frac{-1}{3} - \left({im}^{5} \cdot \frac{1}{60} + \left(im + im\right)\right)\right) \cdot \left(0.5 \cdot \cos re\right)
double f(double re, double im) {
        double r3796038 = 0.5;
        double r3796039 = re;
        double r3796040 = cos(r3796039);
        double r3796041 = r3796038 * r3796040;
        double r3796042 = 0.0;
        double r3796043 = im;
        double r3796044 = r3796042 - r3796043;
        double r3796045 = exp(r3796044);
        double r3796046 = exp(r3796043);
        double r3796047 = r3796045 - r3796046;
        double r3796048 = r3796041 * r3796047;
        return r3796048;
}
↓
double f(double re, double im) {
        double r3796049 = im;
        double r3796050 = r3796049 * r3796049;
        double r3796051 = r3796049 * r3796050;
        double r3796052 = -0.3333333333333333;
        double r3796053 = r3796051 * r3796052;
        double r3796054 = 5.0;
        double r3796055 = pow(r3796049, r3796054);
        double r3796056 = 0.016666666666666666;
        double r3796057 = r3796055 * r3796056;
        double r3796058 = r3796049 + r3796049;
        double r3796059 = r3796057 + r3796058;
        double r3796060 = r3796053 - r3796059;
        double r3796061 = 0.5;
        double r3796062 = re;
        double r3796063 = cos(r3796062);
        double r3796064 = r3796061 * r3796063;
        double r3796065 = r3796060 * r3796064;
        return r3796065;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.0
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019151 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))