Average Error: 58.1 → 0.8
Time: 54.9s
Precision: 64
Internal Precision: 1408
\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\]
↓
\[\left(0.5 \cdot \cos re\right) \cdot \left(-\frac{1}{60} \cdot {im}^{5}\right) + \left(\left(0.5 \cdot \cos re\right) \cdot \left(-2 \cdot im\right) + \left(0.5 \cdot \cos re\right) \cdot \left(-\frac{1}{3} \cdot {im}^{3}\right)\right)\]

Error

Bits error versus re

Bits error versus im

Target

Original58.1
Target0.3
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(\frac{1}{6} \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(\frac{1}{120} \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

  1. Initial program 58.1

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

  2. Taylor expanded around 0 0.8

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

  4. Applied distribute-neg-in0.8

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

  5. Applied distribute-lft-in0.8

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

  7. Applied distribute-neg-in0.8

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

  8. Applied distribute-lft-in0.8

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

Runtime

Time bar (total: 54.9s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(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))))