2sin (example 3.3)

Average Error: 37.0 → 0.5

Time: 17.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.156573731129723388055868311301635498189 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 1.313003327922032909289180230548081681769 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r97863 = x;
        double r97864 = eps;
        double r97865 = r97863 + r97864;
        double r97866 = sin(r97865);
        double r97867 = sin(r97863);
        double r97868 = r97866 - r97867;
        return r97868;
}

↓

double f(double x, double eps) {
        double r97869 = eps;
        double r97870 = -1.1565737311297234e-08;
        bool r97871 = r97869 <= r97870;
        double r97872 = 1.3130033279220329e-17;
        bool r97873 = r97869 <= r97872;
        double r97874 = !r97873;
        bool r97875 = r97871 || r97874;
        double r97876 = x;
        double r97877 = sin(r97876);
        double r97878 = cos(r97869);
        double r97879 = r97877 * r97878;
        double r97880 = cos(r97876);
        double r97881 = sin(r97869);
        double r97882 = r97880 * r97881;
        double r97883 = r97879 + r97882;
        double r97884 = r97883 - r97877;
        double r97885 = 2.0;
        double r97886 = r97869 / r97885;
        double r97887 = sin(r97886);
        double r97888 = r97876 + r97869;
        double r97889 = r97888 + r97876;
        double r97890 = r97889 / r97885;
        double r97891 = cos(r97890);
        double r97892 = r97887 * r97891;
        double r97893 = r97885 * r97892;
        double r97894 = r97875 ? r97884 : r97893;
        return r97894;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.0
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -1.1565737311297234e-08 or 1.3130033279220329e-17 < eps

   1. Initial program 29.8

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

   3. Applied sin-sum 0.8

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

   if -1.1565737311297234e-08 < eps < 1.3130033279220329e-17

   1. Initial program 44.9

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

   3. Applied diff-sin 44.9

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

   4. Simplified 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.156573731129723388055868311301635498189 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 1.313003327922032909289180230548081681769 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))