2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 6.8s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\sin x, \left(\cos \varepsilon - 1\right) + 0, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

C

double f(double x, double eps) {
        double r150474 = x;
        double r150475 = eps;
        double r150476 = r150474 + r150475;
        double r150477 = sin(r150476);
        double r150478 = sin(r150474);
        double r150479 = r150477 - r150478;
        return r150479;
}

↓

double f(double x, double eps) {
        double r150480 = x;
        double r150481 = sin(r150480);
        double r150482 = eps;
        double r150483 = cos(r150482);
        double r150484 = 1.0;
        double r150485 = r150483 - r150484;
        double r150486 = 0.0;
        double r150487 = r150485 + r150486;
        double r150488 = cos(r150480);
        double r150489 = sin(r150482);
        double r150490 = r150488 * r150489;
        double r150491 = fma(r150481, r150487, r150490);
        double r150492 = -r150481;
        double r150493 = r150481 * r150484;
        double r150494 = fma(r150492, r150484, r150493);
        double r150495 = r150491 + r150494;
        return r150495;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	15.1
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 37.2

   \[\sin \left(x + \varepsilon\right) - \sin x\]
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. Applied associate--l+ 22.0

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

6. Applied *-un-lft-identity 22.0

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

7. Applied prod-diff 22.0

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

8. Applied associate-+r+ 22.0

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

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 add-log-exp 0.4

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

12. Applied add-log-exp 0.4

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

13. Applied diff-log 0.5

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

14. Simplified 0.4

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

16. Applied add-cube-cbrt 0.4

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

17. Applied add-sqr-sqrt 15.8

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

18. Applied prod-diff 15.8

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

19. Applied exp-sum 15.8

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

20. Applied log-prod 15.8

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

21. Simplified 0.4

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

22. Simplified 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \left(\cos \varepsilon - 1\right) + \color{blue}{0}, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

23. Final simplification 0.4

    \[\leadsto \mathsf{fma}\left(\sin x, \left(\cos \varepsilon - 1\right) + 0, \cos x \cdot \sin \varepsilon\right) + \mathsf{fma}\left(-\sin x, 1, \sin x \cdot 1\right)\]

Reproduce

herbie shell --seed 2020018 +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)))