Average Error: 43.7 → 0.7
Time: 24.1s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3} - \left(\left(im + im\right) + \left(\frac{1}{60} \cdot {\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)}^{5}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}} \cdot \sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}}\right)\right)\right)\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\frac{-1}{3} \cdot {im}^{3} - \left(\left(im + im\right) + \left(\frac{1}{60} \cdot {\left(\sqrt[3]{im} \cdot \sqrt[3]{im}\right)}^{5}\right) \cdot \left(\sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}} \cdot \sqrt[3]{{\left(\sqrt[3]{im}\right)}^{5}}\right)\right)\right)\right)
double f(double re, double im) {
        double r133556 = 0.5;
        double r133557 = re;
        double r133558 = sin(r133557);
        double r133559 = r133556 * r133558;
        double r133560 = im;
        double r133561 = -r133560;
        double r133562 = exp(r133561);
        double r133563 = exp(r133560);
        double r133564 = r133562 - r133563;
        double r133565 = r133559 * r133564;
        return r133565;
}
↓
double f(double re, double im) {
        double r133566 = 0.5;
        double r133567 = re;
        double r133568 = sin(r133567);
        double r133569 = r133566 * r133568;
        double r133570 = -0.3333333333333333;
        double r133571 = im;
        double r133572 = 3.0;
        double r133573 = pow(r133571, r133572);
        double r133574 = r133570 * r133573;
        double r133575 = r133571 + r133571;
        double r133576 = 0.016666666666666666;
        double r133577 = cbrt(r133571);
        double r133578 = r133577 * r133577;
        double r133579 = 5.0;
        double r133580 = pow(r133578, r133579);
        double r133581 = r133576 * r133580;
        double r133582 = pow(r133577, r133579);
        double r133583 = cbrt(r133582);
        double r133584 = r133583 * r133583;
        double r133585 = r133583 * r133584;
        double r133586 = r133581 * r133585;
        double r133587 = r133575 + r133586;
        double r133588 = r133574 - r133587;
        double r133589 = r133569 * r133588;
        return r133589;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.7
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  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

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

  5. Applied add-cube-cbrt0.7

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

  6. Applied unpow-prod-down0.7

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

  7. Applied associate-*r*0.7

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

  9. Applied add-cube-cbrt0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019196 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))