2sin (example 3.3)

Average Error: 37.2 → 0.4

Time: 17.4s

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 r1373785 = x;
        double r1373786 = eps;
        double r1373787 = r1373785 + r1373786;
        double r1373788 = sin(r1373787);
        double r1373789 = sin(r1373785);
        double r1373790 = r1373788 - r1373789;
        return r1373790;
}

↓

double f(double x, double eps) {
        double r1373791 = eps;
        double r1373792 = -1.7789715079372338e-08;
        bool r1373793 = r1373791 <= r1373792;
        double r1373794 = x;
        double r1373795 = sin(r1373794);
        double r1373796 = cos(r1373791);
        double r1373797 = r1373795 * r1373796;
        double r1373798 = cos(r1373794);
        double r1373799 = sin(r1373791);
        double r1373800 = r1373798 * r1373799;
        double r1373801 = r1373797 + r1373800;
        double r1373802 = r1373801 - r1373795;
        double r1373803 = 1.1933581741857647e-08;
        bool r1373804 = r1373791 <= r1373803;
        double r1373805 = 2.0;
        double r1373806 = 0.5;
        double r1373807 = r1373806 * r1373791;
        double r1373808 = sin(r1373807);
        double r1373809 = r1373794 + r1373791;
        double r1373810 = r1373809 + r1373794;
        double r1373811 = r1373810 / r1373805;
        double r1373812 = cos(r1373811);
        double r1373813 = r1373808 * r1373812;
        double r1373814 = r1373805 * r1373813;
        double r1373815 = r1373804 ? r1373814 : r1373802;
        double r1373816 = r1373793 ? r1373802 : r1373815;
        return r1373816;
}

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)))