Average Error: 43.4 → 0.7
Time: 40.8s
Precision: 64
Internal Precision: 128
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left({im}^{5} \cdot \frac{-1}{60}\right) \cdot \left(\sin re \cdot 0.5\right) + \left((\frac{-1}{3} \cdot \left(im \cdot im\right) + -2)_* \cdot im\right) \cdot \left(\sin re \cdot 0.5\right)\]

Error

Bits error versus re

Bits error versus im

Target

Original43.4
Target0.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.4

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

    \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \color{blue}{(\left({im}^{5}\right) \cdot \frac{-1}{60} + \left((\frac{-1}{3} \cdot \left(im \cdot im\right) + -2)_* \cdot im\right))_*}\]
  4. Using strategy rm

  5. Applied fma-udef0.7

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

  6. Applied distribute-rgt-in0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(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))))