2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 21.2s

Precision: 64

Math

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

↓

\[2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right)}\right)\right)\right)\]

C

double f(double x, double eps) {
        double r5842848 = x;
        double r5842849 = eps;
        double r5842850 = r5842848 + r5842849;
        double r5842851 = sin(r5842850);
        double r5842852 = sin(r5842848);
        double r5842853 = r5842851 - r5842852;
        return r5842853;
}

↓

double f(double x, double eps) {
        double r5842854 = 2.0;
        double r5842855 = eps;
        double r5842856 = 0.5;
        double r5842857 = r5842855 * r5842856;
        double r5842858 = sin(r5842857);
        double r5842859 = x;
        double r5842860 = cos(r5842859);
        double r5842861 = cos(r5842857);
        double r5842862 = r5842860 * r5842861;
        double r5842863 = r5842858 * r5842858;
        double r5842864 = r5842858 * r5842863;
        double r5842865 = sin(r5842859);
        double r5842866 = r5842865 * r5842865;
        double r5842867 = r5842865 * r5842866;
        double r5842868 = r5842864 * r5842867;
        double r5842869 = cbrt(r5842868);
        double r5842870 = r5842862 - r5842869;
        double r5842871 = log1p(r5842870);
        double r5842872 = expm1(r5842871);
        double r5842873 = r5842858 * r5842872;
        double r5842874 = r5842854 * r5842873;
        return r5842874;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	15.0
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 diff-sin 37.2

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

4. Simplified 15.0

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)}\]
5. Using strategy rm

6. Applied expm1-log1p-u 15.1

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\right)}\right)\]

7. Simplified 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\cos \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)\right)}\right)\right)\]
8. Using strategy rm

9. Applied fma-udef 15.0

   \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)}\right)\right)\right)\]

10. Applied cos-sum 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin x}\right)\right)\right)\]
11. Using strategy rm

12. Applied add-cbrt-cube 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \color{blue}{\sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}}\right)\right)\right)\]

13. Applied add-cbrt-cube 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)}} \cdot \sqrt[3]{\left(\sin x \cdot \sin x\right) \cdot \sin x}\right)\right)\right)\]

14. Applied cbrt-unprod 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos x - \color{blue}{\sqrt[3]{\left(\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right) \cdot \left(\left(\sin x \cdot \sin x\right) \cdot \sin x\right)}}\right)\right)\right)\]

15. Final simplification 0.4

    \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot \cos \left(\varepsilon \cdot \frac{1}{2}\right) - \sqrt[3]{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin x \cdot \left(\sin x \cdot \sin x\right)\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

  (- (sin (+ x eps)) (sin x)))