2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 6.2s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin \varepsilon, \cos x, \sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}}\right)\]

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) fma(((double) sin(eps)), ((double) cos(x)), ((double) cbrt(((double) pow(((double) (((double) sin(x)) * ((double) (((double) cos(eps)) - 1.0)))), 3.0))))));
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.9
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 22.0

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

5. Applied associate--l+ 0.4

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

6. Simplified 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied distribute-lft-out 0.4

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

11. Simplified 0.4

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot \left(\cos \varepsilon - 1\right)\right)}\]
12. Using strategy rm

13. Applied add-cbrt-cube 0.4

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

14. Applied add-cbrt-cube 0.5

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

15. Applied cbrt-unprod 0.4

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

16. Simplified 0.4

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

17. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \sqrt[3]{{\left(\sin x \cdot \left(\cos \varepsilon - 1\right)\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(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)))