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

↓

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

Target

Original	43.3
Target	0.3
Herbie	0.8

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

Derivation

Initial program 43.3
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
Taylor expanded around 0 0.8
\[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{\left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\frac{1}{60} \cdot {im}^{5} + 2 \cdot im\right)\right)\right)}\]
Using strategy rm
Applied add-log-exp0.8
\[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(-\left(\frac{1}{3} \cdot {im}^{3} + \left(\color{blue}{\log \left(e^{\frac{1}{60} \cdot {im}^{5}}\right)} + 2 \cdot im\right)\right)\right)\]
Final simplification0.8
\[\leadsto \left(\left(-0.5\right) \cdot \sin re\right) \cdot \left(\left(2 \cdot im + \log \left(e^{{im}^{5} \cdot \frac{1}{60}}\right)\right) + {im}^{3} \cdot \frac{1}{3}\right)\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.8	0.8	0.1	0.8	0%

herbie shell --seed 2018352 
(FPCore (re im)
  :name "math.cos on complex, imaginary part"

  :herbie-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

Try it out

Target

Derivation

Runtime

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