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

↓

\[(\left((\left(im \cdot im\right) \cdot \frac{1}{3} + 2)_*\right) \cdot \left(-im\right) + \left(\left(-\frac{1}{60}\right) \cdot {im}^{5}\right))_* \cdot \left(0.5 \cdot \cos re\right)\]

Target

Original	58.1
Target	0.2
Herbie	0.8

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

Derivation

Initial program 58.1
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
Taylor expanded around 0 0.8
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]
Simplified0.8
\[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{(\left((\left(im \cdot im\right) \cdot \frac{1}{3} + 2)_*\right) \cdot \left(-im\right) + \left(\left(-\frac{1}{60}\right) \cdot {im}^{5}\right))_*}\]
Final simplification0.8
\[\leadsto (\left((\left(im \cdot im\right) \cdot \frac{1}{3} + 2)_*\right) \cdot \left(-im\right) + \left(\left(-\frac{1}{60}\right) \cdot {im}^{5}\right))_* \cdot \left(0.5 \cdot \cos re\right)\]

Runtime

herbie shell --seed 2018217 +o rules:numerics
(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))))

Error

Target

Derivation

Runtime