2sin (example 3.3)

Average Error: 36.8 → 0.4

Time: 18.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -2.754166658325086627164164643222221684482 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 5.156835808069300787421688310116496545277 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r135796 = x;
        double r135797 = eps;
        double r135798 = r135796 + r135797;
        double r135799 = sin(r135798);
        double r135800 = sin(r135796);
        double r135801 = r135799 - r135800;
        return r135801;
}

↓

double f(double x, double eps) {
        double r135802 = eps;
        double r135803 = -2.7541666583250866e-05;
        bool r135804 = r135802 <= r135803;
        double r135805 = 5.156835808069301e-09;
        bool r135806 = r135802 <= r135805;
        double r135807 = !r135806;
        bool r135808 = r135804 || r135807;
        double r135809 = x;
        double r135810 = sin(r135809);
        double r135811 = cos(r135802);
        double r135812 = r135810 * r135811;
        double r135813 = cos(r135809);
        double r135814 = sin(r135802);
        double r135815 = r135813 * r135814;
        double r135816 = r135812 + r135815;
        double r135817 = r135816 - r135810;
        double r135818 = 2.0;
        double r135819 = r135802 / r135818;
        double r135820 = sin(r135819);
        double r135821 = r135809 + r135802;
        double r135822 = r135821 + r135809;
        double r135823 = r135822 / r135818;
        double r135824 = cos(r135823);
        double r135825 = r135820 * r135824;
        double r135826 = r135818 * r135825;
        double r135827 = r135808 ? r135817 : r135826;
        return r135827;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.8
Target	14.8
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 < -2.7541666583250866e-05 or 5.156835808069301e-09 < eps

   1. Initial program 29.0

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

   3. Applied sin-sum 0.5

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

   if -2.7541666583250866e-05 < eps < 5.156835808069301e-09

   1. Initial program 45.0

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

   3. Applied diff-sin 45.0

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

   4. Simplified 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -2.754166658325086627164164643222221684482 \cdot 10^{-5} \lor \neg \left(\varepsilon \le 5.156835808069300787421688310116496545277 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(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)))