2sin (example 3.3)

Average Error: 37.1 → 0.2

Time: 6.3s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin x, -\frac{\sin \varepsilon}{1} \cdot \tan \left(\frac{\varepsilon}{2}\right), \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

C

double f(double x, double eps) {
        double r108479 = x;
        double r108480 = eps;
        double r108481 = r108479 + r108480;
        double r108482 = sin(r108481);
        double r108483 = sin(r108479);
        double r108484 = r108482 - r108483;
        return r108484;
}

↓

double f(double x, double eps) {
        double r108485 = x;
        double r108486 = sin(r108485);
        double r108487 = eps;
        double r108488 = sin(r108487);
        double r108489 = 1.0;
        double r108490 = r108488 / r108489;
        double r108491 = 2.0;
        double r108492 = r108487 / r108491;
        double r108493 = tan(r108492);
        double r108494 = r108490 * r108493;
        double r108495 = -r108494;
        double r108496 = cos(r108485);
        double r108497 = r108496 * r108488;
        double r108498 = fma(r108486, r108495, r108497);
        double r108499 = -r108486;
        double r108500 = fma(r108499, r108489, r108486);
        double r108501 = r108498 + r108500;
        return r108501;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.9
Herbie	0.2

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

Derivation

1. Initial program 37.1

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

3. Applied sin-sum 22.0

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

5. Applied add-cube-cbrt 22.6

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

6. Applied add-sqr-sqrt 43.2

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

7. Applied prod-diff 43.2

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

8. Simplified 22.4

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

9. Simplified 0.4

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

11. Applied flip-- 0.5

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

12. Simplified 0.5

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

14. Applied sub-1-cos 0.3

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

15. Applied distribute-frac-neg 0.3

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

16. Simplified 0.2

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

17. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(\sin x, -\frac{\sin \varepsilon}{1} \cdot \tan \left(\frac{\varepsilon}{2}\right), \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x\right)\]

Reproduce

herbie shell --seed 2019347 +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)))