math.sin on complex, imaginary part

Average Error: 58.0 → 0.8

Time: 24.5s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r115253 = 0.5;
        double r115254 = re;
        double r115255 = cos(r115254);
        double r115256 = r115253 * r115255;
        double r115257 = 0.0;
        double r115258 = im;
        double r115259 = r115257 - r115258;
        double r115260 = exp(r115259);
        double r115261 = exp(r115258);
        double r115262 = r115260 - r115261;
        double r115263 = r115256 * r115262;
        return r115263;
}

↓

double f(double re, double im) {
        double r115264 = re;
        double r115265 = cos(r115264);
        double r115266 = im;
        double r115267 = r115265 * r115266;
        double r115268 = -2.0;
        double r115269 = r115267 * r115268;
        double r115270 = 5.0;
        double r115271 = pow(r115266, r115270);
        double r115272 = 0.016666666666666666;
        double r115273 = r115271 * r115272;
        double r115274 = 3.0;
        double r115275 = pow(r115266, r115274);
        double r115276 = 0.3333333333333333;
        double r115277 = r115275 * r115276;
        double r115278 = r115273 + r115277;
        double r115279 = r115278 * r115265;
        double r115280 = r115269 - r115279;
        double r115281 = 0.5;
        double r115282 = r115280 * r115281;
        return r115282;
}

Error

Bits error versus re

Bits error versus im

Target

Original	58.0
Target	0.3
Herbie	0.8

\[\begin{array}{l} \mathbf{if}\;\left|im\right| \lt 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.1666666666666666574148081281236954964697 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333217685101601546193705872 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0.0 - im} - e^{im}\right)\\ \end{array}\]

Derivation

1. Initial program 58.0

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

2. Simplified 58.0

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

3. Taylor expanded around 0 0.8

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

4. Simplified 0.8

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

6. Applied add-cube-cbrt 1.3

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

7. Applied associate-*r* 1.3

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

8. Simplified 1.3

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

10. Applied pow1/3 16.5

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

11. Applied pow1/3 16.4

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

12. Applied pow-prod-down 1.0

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

13. Simplified 1.0

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

14. Taylor expanded around inf 0.8

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

15. Simplified 0.8

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

16. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019194 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (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))))