2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 15.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7789715079372338 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1933581741857647 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r1970837 = x;
        double r1970838 = eps;
        double r1970839 = r1970837 + r1970838;
        double r1970840 = sin(r1970839);
        double r1970841 = sin(r1970837);
        double r1970842 = r1970840 - r1970841;
        return r1970842;
}

↓

double f(double x, double eps) {
        double r1970843 = eps;
        double r1970844 = -1.7789715079372338e-08;
        bool r1970845 = r1970843 <= r1970844;
        double r1970846 = x;
        double r1970847 = sin(r1970846);
        double r1970848 = cos(r1970843);
        double r1970849 = r1970847 * r1970848;
        double r1970850 = cos(r1970846);
        double r1970851 = sin(r1970843);
        double r1970852 = r1970850 * r1970851;
        double r1970853 = r1970849 + r1970852;
        double r1970854 = r1970853 - r1970847;
        double r1970855 = 1.1933581741857647e-08;
        bool r1970856 = r1970843 <= r1970855;
        double r1970857 = 2.0;
        double r1970858 = 0.5;
        double r1970859 = r1970858 * r1970843;
        double r1970860 = sin(r1970859);
        double r1970861 = r1970846 + r1970843;
        double r1970862 = r1970861 + r1970846;
        double r1970863 = r1970862 / r1970857;
        double r1970864 = cos(r1970863);
        double r1970865 = r1970860 * r1970864;
        double r1970866 = r1970857 * r1970865;
        double r1970867 = r1970856 ? r1970866 : r1970854;
        double r1970868 = r1970845 ? r1970854 : r1970867;
        return r1970868;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.2
Target	15.0
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 < -1.7789715079372338e-08 or 1.1933581741857647e-08 < eps

   1. Initial program 29.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -1.7789715079372338e-08 < eps < 1.1933581741857647e-08

   1. Initial program 45.0

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.7789715079372338 \cdot 10^{-08}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.1933581741857647 \cdot 10^{-08}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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