Average Error: 43.5 → 0.8
Time: 8.9s
Precision: 64
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\left(\sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)}\right) - 2 \cdot im\right)\]
\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)
↓
\left(0.5 \cdot \sin re\right) \cdot \left(\left(-\left(\sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left({im}^{3}, \frac{1}{3}, \left(\sqrt[3]{\frac{1}{60} \cdot {im}^{5}} \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right) \cdot \sqrt[3]{\frac{1}{60} \cdot {im}^{5}}\right)}\right) - 2 \cdot im\right)
double code(double re, double im) {
	return ((0.5 * sin(re)) * (exp(-im) - exp(im)));
}
↓
double code(double re, double im) {
	return ((0.5 * sin(re)) * (-((cbrt(fma(pow(im, 3.0), 0.3333333333333333, ((cbrt((0.016666666666666666 * pow(im, 5.0))) * cbrt((0.016666666666666666 * pow(im, 5.0)))) * cbrt((0.016666666666666666 * pow(im, 5.0)))))) * cbrt(fma(pow(im, 3.0), 0.3333333333333333, ((cbrt((0.016666666666666666 * pow(im, 5.0))) * cbrt((0.016666666666666666 * pow(im, 5.0)))) * cbrt((0.016666666666666666 * pow(im, 5.0))))))) * cbrt(fma(pow(im, 3.0), 0.3333333333333333, ((cbrt((0.016666666666666666 * pow(im, 5.0))) * cbrt((0.016666666666666666 * pow(im, 5.0)))) * cbrt((0.016666666666666666 * pow(im, 5.0))))))) - (2.0 * im)));
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original43.5
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.5

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

  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

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

  5. Applied fma-udef0.8

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

  6. Applied associate--r+0.8

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

  7. Simplified0.8

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

  9. Applied add-cube-cbrt0.8

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

  11. Applied add-cube-cbrt0.8

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

  12. Final simplification0.8

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))