Average Error: 43.7 → 0.9
Time: 43.1s
Precision: 64
Internal Precision: 1344
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
↓
\[\left(\left(-im\right) \cdot \sin re\right) \cdot (0.16666666666666666 \cdot \left(im \cdot im\right) + 1.0)_*\]

Error

Bits error versus re

Bits error versus im

Target

Original43.7
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 43.7

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  4. Taylor expanded around inf 0.9

    \[\leadsto \color{blue}{-\left(0.16666666666666666 \cdot \left(\sin re \cdot {im}^{3}\right) + 1.0 \cdot \left(\sin re \cdot im\right)\right)}\]

  5. Simplified0.9

    \[\leadsto \color{blue}{\left(\sin re \cdot \left(-im\right)\right) \cdot (0.16666666666666666 \cdot \left(im \cdot im\right) + 1.0)_*}\]

  6. Final simplification0.9

    \[\leadsto \left(\left(-im\right) \cdot \sin re\right) \cdot (0.16666666666666666 \cdot \left(im \cdot im\right) + 1.0)_*\]

Runtime

Time bar (total: 43.1s)Debug logProfile

herbie shell --seed 2018217 +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))))