Average Error: 58.2 → 28.8
Time: 40.1s
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(\frac{-1}{3} \cdot {im}^{3} + -2 \cdot im\right) \cdot 0.5\]

Error

Bits error versus re

Bits error versus im

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.2
Target0.2
Herbie28.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  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 31.8

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

  4. Simplified31.8

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

  6. Applied flip-+31.9

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

  8. Applied add-exp-log31.9

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

  9. Applied add-exp-log31.9

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

  10. Applied prod-exp31.9

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

  11. Taylor expanded around inf 28.8

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

  12. Final simplification28.8

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

Runtime

Time bar (total: 40.1s)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))))