2sin (example 3.3)

Average Error: 36.7 → 0.4

Time: 6.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r118802 = x;
        double r118803 = eps;
        double r118804 = r118802 + r118803;
        double r118805 = sin(r118804);
        double r118806 = sin(r118802);
        double r118807 = r118805 - r118806;
        return r118807;
}

↓

double f(double x, double eps) {
        double r118808 = x;
        double r118809 = sin(r118808);
        double r118810 = eps;
        double r118811 = cos(r118810);
        double r118812 = 3.0;
        double r118813 = pow(r118811, r118812);
        double r118814 = 1.0;
        double r118815 = r118813 - r118814;
        double r118816 = r118811 + r118814;
        double r118817 = cbrt(r118816);
        double r118818 = r118817 * r118817;
        double r118819 = r118818 * r118817;
        double r118820 = fma(r118811, r118819, r118814);
        double r118821 = r118815 / r118820;
        double r118822 = cos(r118808);
        double r118823 = sin(r118810);
        double r118824 = r118822 * r118823;
        double r118825 = fma(r118809, r118821, r118824);
        return r118825;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
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 36.7

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

3. Applied sin-sum 21.5

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

4. Taylor expanded around inf 21.5

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

5. Simplified 0.4

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

7. Applied flip3-- 0.4

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

8. Simplified 0.4

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

9. Simplified 0.4

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

11. Applied add-cube-cbrt 0.4

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

12. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020021 +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)))