2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 11.5s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

C

double f(double x, double eps) {
        double r150391 = x;
        double r150392 = eps;
        double r150393 = r150391 + r150392;
        double r150394 = sin(r150393);
        double r150395 = sin(r150391);
        double r150396 = r150394 - r150395;
        return r150396;
}

↓

double f(double x, double eps) {
        double r150397 = eps;
        double r150398 = -8.065146023627676e-09;
        bool r150399 = r150397 <= r150398;
        double r150400 = 3.1297217212623504e-09;
        bool r150401 = r150397 <= r150400;
        double r150402 = !r150401;
        bool r150403 = r150399 || r150402;
        double r150404 = x;
        double r150405 = sin(r150404);
        double r150406 = cos(r150397);
        double r150407 = r150405 * r150406;
        double r150408 = cos(r150404);
        double r150409 = sin(r150397);
        double r150410 = r150408 * r150409;
        double r150411 = r150407 + r150410;
        double r150412 = r150411 - r150405;
        double r150413 = r150404 + r150397;
        double r150414 = r150413 + r150404;
        double r150415 = 2.0;
        double r150416 = r150414 / r150415;
        double r150417 = cos(r150416);
        double r150418 = r150397 / r150415;
        double r150419 = sin(r150418);
        double r150420 = r150417 * r150419;
        double r150421 = r150420 * r150415;
        double r150422 = r150403 ? r150412 : r150421;
        return r150422;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
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 < -8.065146023627676e-09 or 3.1297217212623504e-09 < eps

   1. Initial program 29.9

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

   3. Applied sin-sum 0.6

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

   if -8.065146023627676e-09 < eps < 3.1297217212623504e-09

   1. Initial program 44.8

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

   3. Applied diff-sin 44.8

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

   4. Simplified 0.3

      \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\frac{0 + \varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{\varepsilon}{2}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))