2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 6.5s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.4
Herbie	0.4

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

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 22.5

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

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 0.4

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

11. Applied add-log-exp 0.4

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

12. Applied diff-log 0.4

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

13. Simplified 0.4

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

15. Applied add-log-exp 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020066 
(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)))