2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 23.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r5476806 = x;
        double r5476807 = eps;
        double r5476808 = r5476806 + r5476807;
        double r5476809 = sin(r5476808);
        double r5476810 = sin(r5476806);
        double r5476811 = r5476809 - r5476810;
        return r5476811;
}

↓

double f(double x, double eps) {
        double r5476812 = eps;
        double r5476813 = -5.893930927518443e-09;
        bool r5476814 = r5476812 <= r5476813;
        double r5476815 = x;
        double r5476816 = sin(r5476815);
        double r5476817 = cos(r5476812);
        double r5476818 = r5476816 * r5476817;
        double r5476819 = cos(r5476815);
        double r5476820 = sin(r5476812);
        double r5476821 = r5476819 * r5476820;
        double r5476822 = r5476818 + r5476821;
        double r5476823 = r5476822 - r5476816;
        double r5476824 = 1.8942327691411048e-20;
        bool r5476825 = r5476812 <= r5476824;
        double r5476826 = 2.0;
        double r5476827 = 0.5;
        double r5476828 = r5476812 * r5476827;
        double r5476829 = sin(r5476828);
        double r5476830 = fma(r5476826, r5476815, r5476812);
        double r5476831 = r5476830 / r5476826;
        double r5476832 = cos(r5476831);
        double r5476833 = r5476829 * r5476832;
        double r5476834 = r5476826 * r5476833;
        double r5476835 = r5476825 ? r5476834 : r5476823;
        double r5476836 = r5476814 ? r5476823 : r5476835;
        return r5476836;
}

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

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

4. Final simplification 0.6

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

Reproduce

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

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

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