2sin (example 3.3)

Average Error: 36.5 → 0.5

Time: 6.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r145479 = x;
        double r145480 = eps;
        double r145481 = r145479 + r145480;
        double r145482 = sin(r145481);
        double r145483 = sin(r145479);
        double r145484 = r145482 - r145483;
        return r145484;
}

↓

double f(double x, double eps) {
        double r145485 = x;
        double r145486 = sin(r145485);
        double r145487 = eps;
        double r145488 = cos(r145487);
        double r145489 = 3.0;
        double r145490 = pow(r145488, r145489);
        double r145491 = 1.0;
        double r145492 = r145490 - r145491;
        double r145493 = r145488 + r145491;
        double r145494 = fma(r145488, r145493, r145491);
        double r145495 = r145492 / r145494;
        double r145496 = pow(r145495, r145489);
        double r145497 = cbrt(r145496);
        double r145498 = cos(r145485);
        double r145499 = sin(r145487);
        double r145500 = r145498 * r145499;
        double r145501 = fma(r145486, r145497, r145500);
        double r145502 = -r145486;
        double r145503 = r145486 * r145491;
        double r145504 = fma(r145502, r145491, r145503);
        double r145505 = r145501 + r145504;
        return r145505;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.5
Target	14.8
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 36.5

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

3. Applied sin-sum 21.6

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

4. Applied associate--l+ 21.6

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

6. Applied *-un-lft-identity 21.6

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

7. Applied prod-diff 21.6

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

8. Applied associate-+r+ 21.6

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

9. Simplified 0.4

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

11. Applied flip3-- 0.4

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

12. Simplified 0.4

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

13. Simplified 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \frac{{\left(\cos \varepsilon\right)}^{3} - 1}{\color{blue}{\mathsf{fma}\left(\cos \varepsilon, \cos \varepsilon + 1, 1\right)}}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]
14. Using strategy rm

15. Applied add-cbrt-cube 0.5

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

16. Applied add-cbrt-cube 0.5

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

17. Applied cbrt-undiv 0.5

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

18. Simplified 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019362 +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)))