2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 18.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r4581894 = x;
        double r4581895 = eps;
        double r4581896 = r4581894 + r4581895;
        double r4581897 = sin(r4581896);
        double r4581898 = sin(r4581894);
        double r4581899 = r4581897 - r4581898;
        return r4581899;
}

↓

double f(double x, double eps) {
        double r4581900 = eps;
        double r4581901 = -5.893930927518443e-09;
        bool r4581902 = r4581900 <= r4581901;
        double r4581903 = x;
        double r4581904 = sin(r4581903);
        double r4581905 = cos(r4581900);
        double r4581906 = r4581904 * r4581905;
        double r4581907 = cos(r4581903);
        double r4581908 = sin(r4581900);
        double r4581909 = r4581907 * r4581908;
        double r4581910 = r4581906 + r4581909;
        double r4581911 = r4581910 - r4581904;
        double r4581912 = 1.8942327691411048e-20;
        bool r4581913 = r4581900 <= r4581912;
        double r4581914 = 2.0;
        double r4581915 = fma(r4581914, r4581903, r4581900);
        double r4581916 = r4581915 / r4581914;
        double r4581917 = cos(r4581916);
        double r4581918 = r4581900 / r4581914;
        double r4581919 = sin(r4581918);
        double r4581920 = r4581917 * r4581919;
        double r4581921 = r4581920 * r4581914;
        double r4581922 = r4581913 ? r4581921 : r4581911;
        double r4581923 = r4581902 ? r4581911 : r4581922;
        return r4581923;
}

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

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

4. Final simplification 0.6

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

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)))