2sin (example 3.3)

Average Error: 36.5 → 0.9

Time: 17.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.703827292805686875844406620843923816366 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.449975298893777050395065250516960157615 \cdot 10^{-29}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r102624 = x;
        double r102625 = eps;
        double r102626 = r102624 + r102625;
        double r102627 = sin(r102626);
        double r102628 = sin(r102624);
        double r102629 = r102627 - r102628;
        return r102629;
}

↓

double f(double x, double eps) {
        double r102630 = eps;
        double r102631 = -8.703827292805687e-09;
        bool r102632 = r102630 <= r102631;
        double r102633 = 3.449975298893777e-29;
        bool r102634 = r102630 <= r102633;
        double r102635 = !r102634;
        bool r102636 = r102632 || r102635;
        double r102637 = x;
        double r102638 = sin(r102637);
        double r102639 = cos(r102630);
        double r102640 = r102638 * r102639;
        double r102641 = cos(r102637);
        double r102642 = sin(r102630);
        double r102643 = r102641 * r102642;
        double r102644 = r102640 + r102643;
        double r102645 = r102644 - r102638;
        double r102646 = 2.0;
        double r102647 = r102630 / r102646;
        double r102648 = sin(r102647);
        double r102649 = r102637 + r102630;
        double r102650 = r102649 + r102637;
        double r102651 = r102650 / r102646;
        double r102652 = cos(r102651);
        double r102653 = r102648 * r102652;
        double r102654 = r102646 * r102653;
        double r102655 = r102636 ? r102645 : r102654;
        return r102655;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.5
Target	14.7
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.703827292805687e-09 or 3.449975298893777e-29 < eps

   1. Initial program 29.1

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

   3. Applied sin-sum 1.4

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

   if -8.703827292805687e-09 < eps < 3.449975298893777e-29

   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 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.703827292805686875844406620843923816366 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.449975298893777050395065250516960157615 \cdot 10^{-29}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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)))