2sin (example 3.3)

Average Error: 36.6 → 0.4

Time: 18.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.83922365147160498477619820149701890255 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 7.263195155747593821712365094725644999729 \cdot 10^{-12}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r138907 = x;
        double r138908 = eps;
        double r138909 = r138907 + r138908;
        double r138910 = sin(r138909);
        double r138911 = sin(r138907);
        double r138912 = r138910 - r138911;
        return r138912;
}

↓

double f(double x, double eps) {
        double r138913 = eps;
        double r138914 = -6.839223651471605e-09;
        bool r138915 = r138913 <= r138914;
        double r138916 = 7.263195155747594e-12;
        bool r138917 = r138913 <= r138916;
        double r138918 = !r138917;
        bool r138919 = r138915 || r138918;
        double r138920 = x;
        double r138921 = sin(r138920);
        double r138922 = cos(r138913);
        double r138923 = r138921 * r138922;
        double r138924 = cos(r138920);
        double r138925 = sin(r138913);
        double r138926 = r138924 * r138925;
        double r138927 = r138926 - r138921;
        double r138928 = r138923 + r138927;
        double r138929 = 2.0;
        double r138930 = r138913 / r138929;
        double r138931 = sin(r138930);
        double r138932 = fma(r138929, r138920, r138913);
        double r138933 = r138932 / r138929;
        double r138934 = cos(r138933);
        double r138935 = r138931 * r138934;
        double r138936 = r138929 * r138935;
        double r138937 = r138919 ? r138928 : r138936;
        return r138937;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.6
Target	14.8
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 < -6.839223651471605e-09 or 7.263195155747594e-12 < eps

   1. Initial program 29.8

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

   3. Applied sin-sum 0.6

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

   4. Applied associate--l+ 0.6

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

   if -6.839223651471605e-09 < eps < 7.263195155747594e-12

   1. Initial program 43.7

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

   3. Applied diff-sin 43.7

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

   4. Simplified 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.83922365147160498477619820149701890255 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 7.263195155747593821712365094725644999729 \cdot 10^{-12}\right):\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(2, x, \varepsilon\right)}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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)))