2sin (example 3.3)

Average Error: 37.4 → 0.7

Time: 22.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -109.7910227234083322400692850351333618164 \lor \neg \left(\varepsilon \le 1.164537554863376176631871181292597059554 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r110782 = x;
        double r110783 = eps;
        double r110784 = r110782 + r110783;
        double r110785 = sin(r110784);
        double r110786 = sin(r110782);
        double r110787 = r110785 - r110786;
        return r110787;
}

↓

double f(double x, double eps) {
        double r110788 = eps;
        double r110789 = -109.79102272340833;
        bool r110790 = r110788 <= r110789;
        double r110791 = 1.1645375548633762e-17;
        bool r110792 = r110788 <= r110791;
        double r110793 = !r110792;
        bool r110794 = r110790 || r110793;
        double r110795 = x;
        double r110796 = sin(r110795);
        double r110797 = cos(r110788);
        double r110798 = r110796 * r110797;
        double r110799 = cos(r110795);
        double r110800 = sin(r110788);
        double r110801 = r110799 * r110800;
        double r110802 = r110798 + r110801;
        double r110803 = r110802 - r110796;
        double r110804 = 2.0;
        double r110805 = r110788 / r110804;
        double r110806 = sin(r110805);
        double r110807 = r110795 + r110788;
        double r110808 = r110807 + r110795;
        double r110809 = r110808 / r110804;
        double r110810 = cos(r110809);
        double r110811 = r110806 * r110810;
        double r110812 = r110804 * r110811;
        double r110813 = r110794 ? r110803 : r110812;
        return r110813;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.4
Target	15.3
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if eps < -109.79102272340833 or 1.1645375548633762e-17 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 0.9

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

   if -109.79102272340833 < eps < 1.1645375548633762e-17

   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 0.6

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -109.7910227234083322400692850351333618164 \lor \neg \left(\varepsilon \le 1.164537554863376176631871181292597059554 \cdot 10^{-17}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))