Average Error: 46.9 → 2.5
Time: 15.8s
Precision: 64
Internal precision: 1408
\[\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\]
⬇
\[\left(\left(im + im\right) + \left({im}^{5} \cdot \frac{1}{60} + {im}^3 \cdot \frac{1}{3}\right)\right) \cdot \left(\sin re \cdot \left(-0.5\right)\right)\]

Error

Bits error versus re

Bits error versus im

Target

Original46.9
Comparison11.8
Herbie2.5
\[ \begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\sin re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \end{array} \]

Derivation

  1. Initial program 46.9

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

  2. Applied taylor 2.5

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

  3. Taylor expanded around 0 2.5

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

  4. Applied simplify 2.5

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

  5. Applied taylor 2.5

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

  6. Taylor expanded around 0 2.5

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

  7. Applied simplify 2.5

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

Runtime

Time bar (total: 15.8s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(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))))