2sin (example 3.3)

Average Error: 37.1 → 0.5

Time: 6.0s

Precision: 64

Math

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

↓

\[\sin \varepsilon \cdot \cos x + \frac{\sin x}{\cos \varepsilon + 1} \cdot \frac{\frac{{\left(\cos \varepsilon\right)}^{4}}{1} + \left(-1\right)}{{\left(\cos \varepsilon\right)}^{2} + 1}\]

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(x) / (cos(eps) + 1.0)) * (((pow(cos(eps), 4.0) / 1.0) + -1.0) / (pow(cos(eps), 2.0) + 1.0))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	0.5

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

Derivation

1. Initial program 37.1

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

3. Applied +-commutative 37.1

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

4. Applied sin-sum 21.6

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

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-- 0.6

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

8. Simplified 0.6

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

9. Simplified 0.6

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

11. Applied flip-- 0.6

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

12. Simplified 0.6

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

13. Simplified 0.6

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

15. Applied times-frac 0.6

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

16. Simplified 0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x}{\cos \varepsilon + 1} \cdot \color{blue}{\frac{\frac{{\left(\cos \varepsilon\right)}^{4}}{1} + \left(-1\right)}{{\left(\cos \varepsilon\right)}^{2} + 1}}\]

17. Final simplification 0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \frac{\sin x}{\cos \varepsilon + 1} \cdot \frac{\frac{{\left(\cos \varepsilon\right)}^{4}}{1} + \left(-1\right)}{{\left(\cos \varepsilon\right)}^{2} + 1}\]

Reproduce

herbie shell --seed 2020078 
(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)))