Average Error: 58.0 → 0.8
Time: 28.3s
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 r3684338 = 0.5;
        double r3684339 = re;
        double r3684340 = cos(r3684339);
        double r3684341 = r3684338 * r3684340;
        double r3684342 = 0.0;
        double r3684343 = im;
        double r3684344 = r3684342 - r3684343;
        double r3684345 = exp(r3684344);
        double r3684346 = exp(r3684343);
        double r3684347 = r3684345 - r3684346;
        double r3684348 = r3684341 * r3684347;
        return r3684348;
}
↓
double f(double re, double im) {
        double r3684349 = im;
        double r3684350 = r3684349 * r3684349;
        double r3684351 = r3684349 * r3684350;
        double r3684352 = -0.3333333333333333;
        double r3684353 = r3684351 * r3684352;
        double r3684354 = 5.0;
        double r3684355 = pow(r3684349, r3684354);
        double r3684356 = 0.016666666666666666;
        double r3684357 = r3684355 * r3684356;
        double r3684358 = r3684349 + r3684349;
        double r3684359 = r3684357 + r3684358;
        double r3684360 = r3684353 - r3684359;
        double r3684361 = 0.5;
        double r3684362 = re;
        double r3684363 = cos(r3684362);
        double r3684364 = r3684361 * r3684363;
        double r3684365 = r3684360 * r3684364;
        return r3684365;
}

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
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

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 0.8

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

  3. Simplified0.8

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

  4. Final simplification0.8

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

Reproduce

herbie shell --seed 2019156 
(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))))