2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r116703 = x;
        double r116704 = eps;
        double r116705 = r116703 + r116704;
        double r116706 = sin(r116705);
        double r116707 = sin(r116703);
        double r116708 = r116706 - r116707;
        return r116708;
}

↓

double f(double x, double eps) {
        double r116709 = eps;
        double r116710 = -8.065146023627676e-09;
        bool r116711 = r116709 <= r116710;
        double r116712 = 3.1297217212623504e-09;
        bool r116713 = r116709 <= r116712;
        double r116714 = !r116713;
        bool r116715 = r116711 || r116714;
        double r116716 = x;
        double r116717 = sin(r116716);
        double r116718 = cos(r116709);
        double r116719 = r116717 * r116718;
        double r116720 = cos(r116716);
        double r116721 = sin(r116709);
        double r116722 = r116720 * r116721;
        double r116723 = r116719 + r116722;
        double r116724 = r116723 - r116717;
        double r116725 = 2.0;
        double r116726 = r116709 / r116725;
        double r116727 = sin(r116726);
        double r116728 = r116716 + r116709;
        double r116729 = r116728 + r116716;
        double r116730 = r116729 / r116725;
        double r116731 = cos(r116730);
        double r116732 = r116727 * r116731;
        double r116733 = r116725 * r116732;
        double r116734 = r116715 ? r116724 : r116733;
        return r116734;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
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 2 regimes

2. if eps < -8.065146023627676e-09 or 3.1297217212623504e-09 < eps

   1. Initial program 29.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -8.065146023627676e-09 < eps < 3.1297217212623504e-09

   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 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))