2sin (example 3.3)

Average Error: 37.4 → 0.5

Time: 11.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1161292204334507 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 1.174838876346265 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

C

double f(double x, double eps) {
        double r105753 = x;
        double r105754 = eps;
        double r105755 = r105753 + r105754;
        double r105756 = sin(r105755);
        double r105757 = sin(r105753);
        double r105758 = r105756 - r105757;
        return r105758;
}

↓

double f(double x, double eps) {
        double r105759 = eps;
        double r105760 = -1.1161292204334507e-07;
        bool r105761 = r105759 <= r105760;
        double r105762 = 1.174838876346265e-08;
        bool r105763 = r105759 <= r105762;
        double r105764 = !r105763;
        bool r105765 = r105761 || r105764;
        double r105766 = x;
        double r105767 = sin(r105766);
        double r105768 = cos(r105759);
        double r105769 = r105767 * r105768;
        double r105770 = cos(r105766);
        double r105771 = sin(r105759);
        double r105772 = r105770 * r105771;
        double r105773 = r105769 + r105772;
        double r105774 = r105773 - r105767;
        double r105775 = r105766 + r105759;
        double r105776 = r105775 + r105766;
        double r105777 = 2.0;
        double r105778 = r105776 / r105777;
        double r105779 = cos(r105778);
        double r105780 = r105759 / r105777;
        double r105781 = sin(r105780);
        double r105782 = r105779 * r105781;
        double r105783 = r105782 * r105777;
        double r105784 = r105765 ? r105774 : r105783;
        return r105784;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.1
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 < -1.1161292204334507e-07 or 1.174838876346265e-08 < eps

   1. Initial program 30.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -1.1161292204334507e-07 < eps < 1.174838876346265e-08

   1. Initial program 45.1

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

   3. Applied diff-sin 45.1

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

   4. Simplified 0.4

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.1161292204334507 \cdot 10^{-7} \lor \neg \left(\varepsilon \le 1.174838876346265 \cdot 10^{-8}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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)))