Average Error: 44.0 → 0.8
Time: 32.8s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\sin re \cdot 0.5\right) \cdot \left(-2 \cdot im + \frac{-1}{60} \cdot {im}^{5}\right) + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \cdot \left(\sin re \cdot 0.5\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(\sin re \cdot 0.5\right) \cdot \left(-2 \cdot im + \frac{-1}{60} \cdot {im}^{5}\right) + \left(\left(\left(im \cdot im\right) \cdot im\right) \cdot \frac{-1}{3}\right) \cdot \left(\sin re \cdot 0.5\right)
double f(double re, double im) {
        double r5343634 = 0.5;
        double r5343635 = re;
        double r5343636 = sin(r5343635);
        double r5343637 = r5343634 * r5343636;
        double r5343638 = im;
        double r5343639 = -r5343638;
        double r5343640 = exp(r5343639);
        double r5343641 = exp(r5343638);
        double r5343642 = r5343640 - r5343641;
        double r5343643 = r5343637 * r5343642;
        return r5343643;
}
↓
double f(double re, double im) {
        double r5343644 = re;
        double r5343645 = sin(r5343644);
        double r5343646 = 0.5;
        double r5343647 = r5343645 * r5343646;
        double r5343648 = -2.0;
        double r5343649 = im;
        double r5343650 = r5343648 * r5343649;
        double r5343651 = -0.016666666666666666;
        double r5343652 = 5.0;
        double r5343653 = pow(r5343649, r5343652);
        double r5343654 = r5343651 * r5343653;
        double r5343655 = r5343650 + r5343654;
        double r5343656 = r5343647 * r5343655;
        double r5343657 = r5343649 * r5343649;
        double r5343658 = r5343657 * r5343649;
        double r5343659 = -0.3333333333333333;
        double r5343660 = r5343658 * r5343659;
        double r5343661 = r5343660 * r5343647;
        double r5343662 = r5343656 + r5343661;
        return r5343662;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original44.0
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 44.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied sub-neg0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Simplified0.8

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

  8. Final simplification0.8

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

Reproduce

herbie shell --seed 2019152 
(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))))