2sin (example 3.3)

Average Error: 37.1 → 0.5

Time: 6.5s

Precision: 64

Math

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

↓

\[\sin \varepsilon \cdot \cos x + \sqrt[3]{\cos \varepsilon - 1} \cdot \left(\sin x \cdot \left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right)\right)\]

C

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

↓

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

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.9
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 22.0

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

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 *-un-lft-identity 0.4

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

8. Applied distribute-rgt-out-- 0.4

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

10. Applied add-log-exp 0.5

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

12. Applied add-cube-cbrt 0.6

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

13. Applied associate-*r* 0.6

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

14. Applied exp-prod 0.6

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

15. Applied log-pow 0.6

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

16. Simplified 0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \sqrt[3]{\cos \varepsilon - 1} \cdot \color{blue}{\left(\sin x \cdot \left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right)\right)}\]

17. Final simplification 0.5

    \[\leadsto \sin \varepsilon \cdot \cos x + \sqrt[3]{\cos \varepsilon - 1} \cdot \left(\sin x \cdot \left(\sqrt[3]{\cos \varepsilon - 1} \cdot \sqrt[3]{\cos \varepsilon - 1}\right)\right)\]

Reproduce

herbie shell --seed 2020113 
(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)))