Average Error: 43.6 → 0.7
Time: 30.5s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(\left(\left(\frac{-1}{3} \cdot im\right) \cdot im - 2\right) \cdot im - {im}^{5} \cdot \frac{1}{60}\right) \cdot 0.5\right) \cdot \sin re\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\left(\left(\left(\frac{-1}{3} \cdot im\right) \cdot im - 2\right) \cdot im - {im}^{5} \cdot \frac{1}{60}\right) \cdot 0.5\right) \cdot \sin re
double f(double re, double im) {
        double r11610832 = 0.5;
        double r11610833 = re;
        double r11610834 = sin(r11610833);
        double r11610835 = r11610832 * r11610834;
        double r11610836 = im;
        double r11610837 = -r11610836;
        double r11610838 = exp(r11610837);
        double r11610839 = exp(r11610836);
        double r11610840 = r11610838 - r11610839;
        double r11610841 = r11610835 * r11610840;
        return r11610841;
}
↓
double f(double re, double im) {
        double r11610842 = -0.3333333333333333;
        double r11610843 = im;
        double r11610844 = r11610842 * r11610843;
        double r11610845 = r11610844 * r11610843;
        double r11610846 = 2.0;
        double r11610847 = r11610845 - r11610846;
        double r11610848 = r11610847 * r11610843;
        double r11610849 = 5.0;
        double r11610850 = pow(r11610843, r11610849);
        double r11610851 = 0.016666666666666666;
        double r11610852 = r11610850 * r11610851;
        double r11610853 = r11610848 - r11610852;
        double r11610854 = 0.5;
        double r11610855 = r11610853 * r11610854;
        double r11610856 = re;
        double r11610857 = sin(r11610856);
        double r11610858 = r11610855 * r11610857;
        return r11610858;
}

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

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 0.7

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

  3. Simplified0.7

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

  5. Applied pow10.7

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

  6. Applied pow10.7

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

  7. Applied pow10.7

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

  8. Applied pow-prod-down0.7

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

  9. Applied pow-prod-down0.7

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

  10. Simplified0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

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