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

⬇

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

Target

Original	46.7
Comparison	11.9
Herbie	2.6

\[ \begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Derivation

Initial program 46.7
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
Applied taylor 2.6
\[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\left(2 \cdot im + \left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right)\right)\right)\]
Taylor expanded around 0 2.6

\[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(2 \cdot im + \left(\frac{1}{3} \cdot {im}^{3} + \frac{1}{60} \cdot {im}^{5}\right)\right)\right)}\]
Applied simplify 2.6
\[\leadsto \color{blue}{\left(\left(-0.5\right) \cdot \sin re\right) \cdot \left(\left(\left(im + im\right) + {im}^3 \cdot \frac{1}{3}\right) + {im}^{5} \cdot \frac{1}{60}\right)}\]
Removed slow pow expressions

Runtime

Please include this information when filing a bug report:

herbie --seed '#(3968841166 4289395207 4220785236 3061720342 1096712026 1693760750)'
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

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

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

Error

Target

Derivation

Runtime