Average Error: 43.3 → 0.7
Time: 8.2s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(1 \cdot \left(\left(0.5 \cdot \sin re\right) \cdot {im}^{3}\right)\right) \cdot \frac{-1}{3} + \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(1 \cdot \left(\left(0.5 \cdot \sin re\right) \cdot {im}^{3}\right)\right) \cdot \frac{-1}{3} + \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)
double f(double re, double im) {
        double r181634 = 0.5;
        double r181635 = re;
        double r181636 = sin(r181635);
        double r181637 = r181634 * r181636;
        double r181638 = im;
        double r181639 = -r181638;
        double r181640 = exp(r181639);
        double r181641 = exp(r181638);
        double r181642 = r181640 - r181641;
        double r181643 = r181637 * r181642;
        return r181643;
}
↓
double f(double re, double im) {
        double r181644 = 1.0;
        double r181645 = 0.5;
        double r181646 = re;
        double r181647 = sin(r181646);
        double r181648 = r181645 * r181647;
        double r181649 = im;
        double r181650 = 3.0;
        double r181651 = pow(r181649, r181650);
        double r181652 = r181648 * r181651;
        double r181653 = r181644 * r181652;
        double r181654 = -0.3333333333333333;
        double r181655 = r181653 * r181654;
        double r181656 = 0.016666666666666666;
        double r181657 = 5.0;
        double r181658 = pow(r181649, r181657);
        double r181659 = r181656 * r181658;
        double r181660 = 2.0;
        double r181661 = r181660 * r181649;
        double r181662 = r181659 + r181661;
        double r181663 = -r181662;
        double r181664 = r181648 * r181663;
        double r181665 = r181655 + r181664;
        return r181665;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.3
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.3

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

  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. Using strategy rm

  4. Applied distribute-neg-in0.7

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

  5. Applied distribute-lft-in0.7

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

  6. Simplified0.7

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

  8. Applied *-un-lft-identity0.7

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

  9. Final simplification0.7

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

Reproduce

herbie shell --seed 2020046 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1) (- (* (sin re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (sin re)) (- (exp (- im)) (exp im))))

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