2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 18.9s

Precision: 64

Math

\[\sin \left(x + \varepsilon\right) - \sin x\]

↓

\[2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)\]

C

double f(double x, double eps) {
        double r4228904 = x;
        double r4228905 = eps;
        double r4228906 = r4228904 + r4228905;
        double r4228907 = sin(r4228906);
        double r4228908 = sin(r4228904);
        double r4228909 = r4228907 - r4228908;
        return r4228909;
}

↓

double f(double x, double eps) {
        double r4228910 = 2.0;
        double r4228911 = eps;
        double r4228912 = r4228911 / r4228910;
        double r4228913 = sin(r4228912);
        double r4228914 = 0.5;
        double r4228915 = r4228914 * r4228911;
        double r4228916 = cos(r4228915);
        double r4228917 = x;
        double r4228918 = cos(r4228917);
        double r4228919 = r4228916 * r4228918;
        double r4228920 = sin(r4228915);
        double r4228921 = sin(r4228917);
        double r4228922 = r4228920 * r4228921;
        double r4228923 = r4228919 - r4228922;
        double r4228924 = r4228913 * r4228923;
        double r4228925 = r4228910 * r4228924;
        return r4228925;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	0.3

\[2 \cdot \left(\cos \left(x + \frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\]

Derivation

1. Initial program 37.1

   \[\sin \left(x + \varepsilon\right) - \sin x\]
2. Using strategy rm

3. Applied diff-sin 37.5

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]

4. Simplified 15.3

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
5. Using strategy rm

6. Applied add-log-exp 15.5

   \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}\right)}\right)\]

7. Taylor expanded around inf 15.5

   \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\color{blue}{\cos \left(\frac{1}{2} \cdot \left(2 \cdot x + \varepsilon\right)\right)}}\right)\right)\]

8. Simplified 15.5

   \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\color{blue}{\cos \left(x + \varepsilon \cdot \frac{1}{2}\right)}}\right)\right)\]
9. Using strategy rm

10. Applied cos-sum 0.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \left(e^{\color{blue}{\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}}\right)\right)\]

11. Applied exp-diff 0.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \log \color{blue}{\left(\frac{e^{\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)}}{e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}}\right)}\right)\]

12. Applied log-div 0.6

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\left(\log \left(e^{\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)}\right) - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right)}\right)\]

13. Simplified 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\color{blue}{\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right)} - \log \left(e^{\sin x \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)}\right)\right)\right)\]

14. Simplified 0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin x}\right)\right)\]

15. Final simplification 0.3

    \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x\right)\right)\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  (- (sin (+ x eps)) (sin x)))