2sin (example 3.3)

Average Error: 36.7 → 0.5

Time: 15.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.765190492196658 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r2217710 = x;
        double r2217711 = eps;
        double r2217712 = r2217710 + r2217711;
        double r2217713 = sin(r2217712);
        double r2217714 = sin(r2217710);
        double r2217715 = r2217713 - r2217714;
        return r2217715;
}

↓

double f(double x, double eps) {
        double r2217716 = eps;
        double r2217717 = -0.22306452923435388;
        bool r2217718 = r2217716 <= r2217717;
        double r2217719 = x;
        double r2217720 = sin(r2217719);
        double r2217721 = cos(r2217716);
        double r2217722 = r2217720 * r2217721;
        double r2217723 = cos(r2217719);
        double r2217724 = sin(r2217716);
        double r2217725 = r2217723 * r2217724;
        double r2217726 = r2217722 + r2217725;
        double r2217727 = r2217726 - r2217720;
        double r2217728 = 6.765190492196658e-13;
        bool r2217729 = r2217716 <= r2217728;
        double r2217730 = 2.0;
        double r2217731 = 0.5;
        double r2217732 = r2217731 * r2217716;
        double r2217733 = sin(r2217732);
        double r2217734 = r2217719 + r2217716;
        double r2217735 = r2217734 + r2217719;
        double r2217736 = r2217735 / r2217730;
        double r2217737 = cos(r2217736);
        double r2217738 = r2217733 * r2217737;
        double r2217739 = r2217730 * r2217738;
        double r2217740 = r2217729 ? r2217739 : r2217727;
        double r2217741 = r2217718 ? r2217727 : r2217740;
        return r2217741;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.7
Target	14.5
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 < -0.22306452923435388 or 6.765190492196658e-13 < eps

   1. Initial program 29.3

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -0.22306452923435388 < eps < 6.765190492196658e-13

   1. Initial program 44.2

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

   3. Applied diff-sin 44.2

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

   4. Simplified 0.4

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.765190492196658 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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