2sin (example 3.3)

Average Error: 37.0 → 0.4

Time: 6.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r103836 = x;
        double r103837 = eps;
        double r103838 = r103836 + r103837;
        double r103839 = sin(r103838);
        double r103840 = sin(r103836);
        double r103841 = r103839 - r103840;
        return r103841;
}

↓

double f(double x, double eps) {
        double r103842 = x;
        double r103843 = sin(r103842);
        double r103844 = eps;
        double r103845 = cos(r103844);
        double r103846 = 1.0;
        double r103847 = r103845 - r103846;
        double r103848 = cos(r103842);
        double r103849 = sin(r103844);
        double r103850 = r103848 * r103849;
        double r103851 = fma(r103843, r103847, r103850);
        double r103852 = -r103843;
        double r103853 = r103843 * r103846;
        double r103854 = fma(r103852, r103846, r103853);
        double r103855 = r103851 + r103854;
        return r103855;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
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 37.0

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

3. Applied sin-sum 21.8

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

4. Applied associate--l+ 21.8

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

6. Applied *-un-lft-identity 21.8

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

7. Applied prod-diff 21.8

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

8. Applied associate-+r+ 21.8

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

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 exp-diff 0.5

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

17. Applied log-div 0.4

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

18. Simplified 0.4

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

19. Simplified 0.4

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

20. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020027 +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)))