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Target

Derivation

1. Split input into 2 regimes

2. if (cos re) < 0.9996821205382268

   1. Initial program 58.2

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

   2. Initial simplification58.3

      \[\leadsto 0.5 \cdot \left(\frac{\cos re}{e^{im}} - e^{im} \cdot \cos re\right)\]
   3. Using strategy rm

   4. Applied flip3--58.3

      \[\leadsto 0.5 \cdot \color{blue}{\frac{{\left(\frac{\cos re}{e^{im}}\right)}^{3} - {\left(e^{im} \cdot \cos re\right)}^{3}}{\frac{\cos re}{e^{im}} \cdot \frac{\cos re}{e^{im}} + \left(\left(e^{im} \cdot \cos re\right) \cdot \left(e^{im} \cdot \cos re\right) + \frac{\cos re}{e^{im}} \cdot \left(e^{im} \cdot \cos re\right)\right)}}\]

   if 0.9996821205382268 < (cos re)

   1. Initial program 58.2

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]

   2. Initial simplification58.3

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

   3. Taylor expanded around 0 1.3

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

   4. Simplified1.3

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

   6. Applied flip-+1.3

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

   8. Applied add-exp-log1.3

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

   9. Applied add-exp-log1.3

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

   10. Applied prod-exp1.3

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right) + \frac{\color{blue}{e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)}} - -2 \cdot -2}{re \cdot re - -2}\right) \cdot im\right)\]
   11. Using strategy rm

   12. Applied flip3--1.3

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

   13. Applied associate-/l/1.3

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

   14. Simplified1.3

       \[\leadsto 0.5 \cdot \left(\left(im \cdot \left(im \cdot \frac{-1}{3}\right) + \frac{\color{blue}{-64 + {\left({re}^{4}\right)}^{3}}}{\left(re \cdot re - -2\right) \cdot \left(e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)} \cdot e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)} + \left(\left(-2 \cdot -2\right) \cdot \left(-2 \cdot -2\right) + e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)} \cdot \left(-2 \cdot -2\right)\right)\right)}\right) \cdot im\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification29.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos re \le 0.9996821205382268:\\ \;\;\;\;0.5 \cdot \frac{{\left(\frac{\cos re}{e^{im}}\right)}^{3} - {\left(e^{im} \cdot \cos re\right)}^{3}}{\frac{\cos re}{e^{im}} \cdot \frac{\cos re}{e^{im}} + \left(\left(e^{im} \cdot \cos re\right) \cdot \left(e^{im} \cdot \cos re\right) + \frac{\cos re}{e^{im}} \cdot \left(e^{im} \cdot \cos re\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot \left(im \cdot \left(im \cdot \frac{-1}{3}\right) + \frac{{\left({re}^{4}\right)}^{3} + -64}{\left(\left(16 + e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)} \cdot 4\right) + e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)} \cdot e^{\log \left(re \cdot re\right) + \log \left(re \cdot re\right)}\right) \cdot \left(re \cdot re - -2\right)}\right)\right)\\ \end{array}\]

Runtime

Time bar (total: 36.3s)Debug logProfile

herbie shell --seed 2018255 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1) (- (* (cos re) (+ (+ im (* (* (* 1/6 im) im) im)) (* (* (* (* (* 1/120 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0 im)) (exp im))))