Average Error: 58.0 → 0.7
Time: 24.0s
Precision: 64
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)
↓
\left(0.5 \cdot \cos re\right) \cdot \left(-\left(\left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right) + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r100088 = 0.5;
        double r100089 = re;
        double r100090 = cos(r100089);
        double r100091 = r100088 * r100090;
        double r100092 = 0.0;
        double r100093 = im;
        double r100094 = r100092 - r100093;
        double r100095 = exp(r100094);
        double r100096 = exp(r100093);
        double r100097 = r100095 - r100096;
        double r100098 = r100091 * r100097;
        return r100098;
}
↓
double f(double re, double im) {
        double r100099 = 0.5;
        double r100100 = re;
        double r100101 = cos(r100100);
        double r100102 = r100099 * r100101;
        double r100103 = 0.3333333333333333;
        double r100104 = im;
        double r100105 = 3.0;
        double r100106 = pow(r100104, r100105);
        double r100107 = r100103 * r100106;
        double r100108 = 0.016666666666666666;
        double r100109 = 5.0;
        double r100110 = pow(r100104, r100109);
        double r100111 = r100108 * r100110;
        double r100112 = r100107 + r100111;
        double r100113 = 2.0;
        double r100114 = r100113 * r100104;
        double r100115 = r100112 + r100114;
        double r100116 = -r100115;
        double r100117 = r100102 * r100116;
        return r100117;
}

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
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.0

    \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

  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. Using strategy rm

  4. Applied associate-+r+0.7

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

  5. Final simplification0.7

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

Reproduce

herbie shell --seed 2019297 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 0.166666666666666657 im) im) im)) (* (* (* (* (* 0.00833333333333333322 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))