math.sin on complex, imaginary part

Average Error: 57.9 → 0.7

Time: 15.1s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

↓

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

C

double code(double re, double im) {
	return ((double) (((double) (0.5 * ((double) cos(re)))) * ((double) (((double) exp(((double) (0.0 - im)))) - ((double) exp(im))))));
}

↓

double code(double re, double im) {
	return ((double) (((double) (((double) (-2.0 * im)) * ((double) (0.5 * ((double) cos(re)))))) + ((double) (((double) (0.5 * ((double) cos(re)))) * ((double) (((double) (((double) pow(im, 3.0)) * -0.3333333333333333)) + ((double) (((double) pow(im, 5.0)) * -0.016666666666666666))))))));
}

Error

Bits error versus re

Bits error versus im

Target

Original	57.9
Target	0.3
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.166666666666666657 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.00833333333333333322 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 57.9

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\]

2. Taylor expanded around 0 0.7

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

3. Simplified 0.7

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

5. Applied sub-neg 0.7

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

6. Applied distribute-lft-in 0.7

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

7. Simplified 0.7

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

8. Simplified 0.7

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

9. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020182 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (neg (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))