2sin (example 3.3)

Average Error: 36.7 → 0.3

Time: 5.3s

Precision: 64

Math

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

↓

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

C

double code(double x, double eps) {
	return (sin((x + eps)) - sin(x));
}

↓

double code(double x, double eps) {
	return ((sin(eps) * cos(x)) + (((sin(eps) * sin(eps)) * sin(x)) / -(1.0 + cos(eps))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.9
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 36.7

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

3. Applied +-commutative 36.7

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

4. Applied sin-sum 21.7

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

5. Applied associate--l+ 0.4

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

7. Applied flip-- 0.5

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

8. Simplified 0.5

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

9. Simplified 0.5

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

11. Applied *-commutative 0.5

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

12. Applied *-commutative 0.5

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

13. Applied times-frac 0.5

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

14. Simplified 0.5

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

15. Simplified 0.5

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

17. Applied frac-2neg 0.5

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

18. Applied associate-*r/ 0.5

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

19. Simplified 0.3

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

20. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020071 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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