2sin (example 3.3)

Average Error: 37.1 → 0.8

Time: 18.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -40694913.773095824:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.386517640886532 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r2248843 = x;
        double r2248844 = eps;
        double r2248845 = r2248843 + r2248844;
        double r2248846 = sin(r2248845);
        double r2248847 = sin(r2248843);
        double r2248848 = r2248846 - r2248847;
        return r2248848;
}

↓

double f(double x, double eps) {
        double r2248849 = eps;
        double r2248850 = -40694913.773095824;
        bool r2248851 = r2248849 <= r2248850;
        double r2248852 = x;
        double r2248853 = sin(r2248852);
        double r2248854 = cos(r2248849);
        double r2248855 = r2248853 * r2248854;
        double r2248856 = cos(r2248852);
        double r2248857 = sin(r2248849);
        double r2248858 = r2248856 * r2248857;
        double r2248859 = r2248855 + r2248858;
        double r2248860 = r2248859 - r2248853;
        double r2248861 = 6.386517640886532e-16;
        bool r2248862 = r2248849 <= r2248861;
        double r2248863 = 2.0;
        double r2248864 = r2248849 / r2248863;
        double r2248865 = sin(r2248864);
        double r2248866 = r2248852 + r2248849;
        double r2248867 = r2248852 + r2248866;
        double r2248868 = r2248867 / r2248863;
        double r2248869 = cos(r2248868);
        double r2248870 = r2248865 * r2248869;
        double r2248871 = r2248863 * r2248870;
        double r2248872 = r2248862 ? r2248871 : r2248860;
        double r2248873 = r2248851 ? r2248860 : r2248872;
        return r2248873;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if eps < -40694913.773095824 or 6.386517640886532e-16 < eps

   1. Initial program 29.7

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

   3. Applied sin-sum 0.6

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

   if -40694913.773095824 < eps < 6.386517640886532e-16

   1. Initial program 44.8

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

   3. Applied diff-sin 44.8

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

   4. Simplified 0.9

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -40694913.773095824:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.386517640886532 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x eps)
  :name "2sin (example 3.3)"

  :herbie-target
  (* 2 (* (cos (+ x (/ eps 2))) (sin (/ eps 2))))

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