2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 20.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r78844 = x;
        double r78845 = eps;
        double r78846 = r78844 + r78845;
        double r78847 = sin(r78846);
        double r78848 = sin(r78844);
        double r78849 = r78847 - r78848;
        return r78849;
}

↓

double f(double x, double eps) {
        double r78850 = eps;
        double r78851 = -9.760584355965869e-05;
        bool r78852 = r78850 <= r78851;
        double r78853 = x;
        double r78854 = sin(r78853);
        double r78855 = cos(r78850);
        double r78856 = r78854 * r78855;
        double r78857 = cos(r78853);
        double r78858 = sin(r78850);
        double r78859 = r78857 * r78858;
        double r78860 = r78856 + r78859;
        double r78861 = r78860 - r78854;
        double r78862 = 9.433852813731428e-09;
        bool r78863 = r78850 <= r78862;
        double r78864 = r78853 + r78850;
        double r78865 = r78864 + r78853;
        double r78866 = 2.0;
        double r78867 = r78865 / r78866;
        double r78868 = cos(r78867);
        double r78869 = r78850 / r78866;
        double r78870 = sin(r78869);
        double r78871 = r78870 * r78866;
        double r78872 = r78868 * r78871;
        double r78873 = r78859 - r78854;
        double r78874 = r78856 + r78873;
        double r78875 = r78863 ? r78872 : r78874;
        double r78876 = r78852 ? r78861 : r78875;
        return r78876;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.8
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if eps < -9.760584355965869e-05

   1. Initial program 30.4

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

   3. Applied sin-sum 0.4

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

   if -9.760584355965869e-05 < eps < 9.433852813731428e-09

   1. Initial program 44.6

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

   3. Applied diff-sin 44.6

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

   4. Simplified 0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied pow1 0.3

      \[\leadsto 2 \cdot \left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \color{blue}{{\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}^{1}}\right)\]

   7. Applied pow1 0.3

      \[\leadsto 2 \cdot \left(\color{blue}{{\left(\sin \left(\frac{\varepsilon - 0}{2}\right)\right)}^{1}} \cdot {\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}^{1}\right)\]

   8. Applied pow-prod-down 0.3

      \[\leadsto 2 \cdot \color{blue}{{\left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}^{1}}\]

   9. Applied pow1 0.3

      \[\leadsto \color{blue}{{2}^{1}} \cdot {\left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}^{1}\]

   10. Applied pow-prod-down 0.3

       \[\leadsto \color{blue}{{\left(2 \cdot \left(\sin \left(\frac{\varepsilon - 0}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)}^{1}}\]

   11. Simplified 0.3

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

   if 9.433852813731428e-09 < eps

   1. Initial program 28.2

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

   3. Applied sin-sum 0.5

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

   4. Applied associate--l+ 0.5

      \[\leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -9.760584355965868752318115308952428677003 \cdot 10^{-5}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 9.433852813731427826173936825124993754699 \cdot 10^{-9}:\\ \;\;\;\;\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))