2sin (example 3.3)

Average Error: 37.1 → 0.3

Time: 21.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r4788662 = x;
        double r4788663 = eps;
        double r4788664 = r4788662 + r4788663;
        double r4788665 = sin(r4788664);
        double r4788666 = sin(r4788662);
        double r4788667 = r4788665 - r4788666;
        return r4788667;
}

↓

double f(double x, double eps) {
        double r4788668 = 2.0;
        double r4788669 = x;
        double r4788670 = cos(r4788669);
        double r4788671 = 0.5;
        double r4788672 = eps;
        double r4788673 = r4788671 * r4788672;
        double r4788674 = cos(r4788673);
        double r4788675 = r4788670 * r4788674;
        double r4788676 = sin(r4788673);
        double r4788677 = sin(r4788669);
        double r4788678 = r4788676 * r4788677;
        double r4788679 = r4788675 - r4788678;
        double r4788680 = r4788679 * r4788676;
        double r4788681 = r4788668 * r4788680;
        return r4788681;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.0
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 15.1

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

6. Applied add-cbrt-cube 15.2

   \[\leadsto 2 \cdot \left(\color{blue}{\sqrt[3]{\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]
7. Using strategy rm

8. Applied add-log-exp 15.4

   \[\leadsto 2 \cdot \left(\color{blue}{\log \left(e^{\sqrt[3]{\left(\cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)}}\right)} \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)\]

9. Simplified 15.2

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

11. Applied cos-sum 0.6

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

12. Applied exp-diff 0.6

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

13. Applied log-div 0.6

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

14. Simplified 0.4

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019171 
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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