2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 19.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -5.893930927518443 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.8942327691411048 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r5731845 = x;
        double r5731846 = eps;
        double r5731847 = r5731845 + r5731846;
        double r5731848 = sin(r5731847);
        double r5731849 = sin(r5731845);
        double r5731850 = r5731848 - r5731849;
        return r5731850;
}

↓

double f(double x, double eps) {
        double r5731851 = eps;
        double r5731852 = -5.893930927518443e-09;
        bool r5731853 = r5731851 <= r5731852;
        double r5731854 = x;
        double r5731855 = sin(r5731854);
        double r5731856 = cos(r5731851);
        double r5731857 = r5731855 * r5731856;
        double r5731858 = cos(r5731854);
        double r5731859 = sin(r5731851);
        double r5731860 = r5731858 * r5731859;
        double r5731861 = r5731857 + r5731860;
        double r5731862 = r5731861 - r5731855;
        double r5731863 = 1.8942327691411048e-20;
        bool r5731864 = r5731851 <= r5731863;
        double r5731865 = 2.0;
        double r5731866 = r5731854 + r5731851;
        double r5731867 = r5731866 + r5731854;
        double r5731868 = r5731867 / r5731865;
        double r5731869 = cos(r5731868);
        double r5731870 = 0.5;
        double r5731871 = r5731851 * r5731870;
        double r5731872 = sin(r5731871);
        double r5731873 = r5731869 * r5731872;
        double r5731874 = r5731865 * r5731873;
        double r5731875 = r5731864 ? r5731874 : r5731862;
        double r5731876 = r5731853 ? r5731862 : r5731875;
        return r5731876;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.9
Target	15.3
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if eps < -5.893930927518443e-09 or 1.8942327691411048e-20 < eps

   1. Initial program 30.1

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

   3. Applied sin-sum 1.0

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

   if -5.893930927518443e-09 < eps < 1.8942327691411048e-20

   1. Initial program 44.5

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

   3. Applied diff-sin 44.5

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

   4. Simplified 0.2

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -5.893930927518443 \cdot 10^{-09}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 1.8942327691411048 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\varepsilon \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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