Average Error: 43.8 → 0.7
Time: 59.3s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60} - im \cdot \left(2 + im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left({im}^{5} \cdot \frac{-1}{60} - im \cdot \left(2 + im \cdot \left(\frac{1}{3} \cdot im\right)\right)\right) \cdot \left(0.5 \cdot \sin re\right)
double f(double re, double im) {
        double r41024447 = 0.5;
        double r41024448 = re;
        double r41024449 = sin(r41024448);
        double r41024450 = r41024447 * r41024449;
        double r41024451 = im;
        double r41024452 = -r41024451;
        double r41024453 = exp(r41024452);
        double r41024454 = exp(r41024451);
        double r41024455 = r41024453 - r41024454;
        double r41024456 = r41024450 * r41024455;
        return r41024456;
}
↓
double f(double re, double im) {
        double r41024457 = im;
        double r41024458 = 5.0;
        double r41024459 = pow(r41024457, r41024458);
        double r41024460 = -0.016666666666666666;
        double r41024461 = r41024459 * r41024460;
        double r41024462 = 2.0;
        double r41024463 = 0.3333333333333333;
        double r41024464 = r41024463 * r41024457;
        double r41024465 = r41024457 * r41024464;
        double r41024466 = r41024462 + r41024465;
        double r41024467 = r41024457 * r41024466;
        double r41024468 = r41024461 - r41024467;
        double r41024469 = 0.5;
        double r41024470 = re;
        double r41024471 = sin(r41024470);
        double r41024472 = r41024469 * r41024471;
        double r41024473 = r41024468 * r41024472;
        return r41024473;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.8
Target0.2
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 43.8

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

  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

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019121 
(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))))