2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 6.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r116810 = x;
        double r116811 = eps;
        double r116812 = r116810 + r116811;
        double r116813 = sin(r116812);
        double r116814 = sin(r116810);
        double r116815 = r116813 - r116814;
        return r116815;
}

↓

double f(double x, double eps) {
        double r116816 = x;
        double r116817 = sin(r116816);
        double r116818 = eps;
        double r116819 = cos(r116818);
        double r116820 = r116819 * r116819;
        double r116821 = 1.0;
        double r116822 = -r116821;
        double r116823 = fma(r116819, r116820, r116822);
        double r116824 = r116819 + r116821;
        double r116825 = fma(r116819, r116824, r116821);
        double r116826 = r116823 / r116825;
        double r116827 = cos(r116816);
        double r116828 = sin(r116818);
        double r116829 = r116827 * r116828;
        double r116830 = fma(r116817, r116826, r116829);
        return r116830;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
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 36.8

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

3. Applied sin-sum 21.7

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

5. Applied *-un-lft-identity 21.7

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

6. Applied *-un-lft-identity 21.7

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

7. Applied distribute-lft-out-- 21.7

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

8. Simplified 0.4

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

10. Applied flip3-- 0.4

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

11. Simplified 0.4

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

12. Simplified 0.4

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

14. Applied cube-mult 0.4

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

15. Applied fma-neg 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020089 +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)))