2sin (example 3.3)

Average Error: 36.5 → 0.5

Time: 14.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.131893915803982637690050267343516710028 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.144339914669044167378723460572493433372 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r123282 = x;
        double r123283 = eps;
        double r123284 = r123282 + r123283;
        double r123285 = sin(r123284);
        double r123286 = sin(r123282);
        double r123287 = r123285 - r123286;
        return r123287;
}

↓

double f(double x, double eps) {
        double r123288 = eps;
        double r123289 = -8.131893915803983e-09;
        bool r123290 = r123288 <= r123289;
        double r123291 = 3.144339914669044e-18;
        bool r123292 = r123288 <= r123291;
        double r123293 = !r123292;
        bool r123294 = r123290 || r123293;
        double r123295 = x;
        double r123296 = sin(r123295);
        double r123297 = cos(r123288);
        double r123298 = r123296 * r123297;
        double r123299 = cos(r123295);
        double r123300 = sin(r123288);
        double r123301 = r123299 * r123300;
        double r123302 = r123298 + r123301;
        double r123303 = r123302 - r123296;
        double r123304 = 2.0;
        double r123305 = r123288 / r123304;
        double r123306 = sin(r123305);
        double r123307 = r123295 + r123288;
        double r123308 = r123307 + r123295;
        double r123309 = r123308 / r123304;
        double r123310 = cos(r123309);
        double r123311 = r123306 * r123310;
        double r123312 = r123304 * r123311;
        double r123313 = r123294 ? r123303 : r123312;
        return r123313;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	36.5
Target	14.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if eps < -8.131893915803983e-09 or 3.144339914669044e-18 < eps

   1. Initial program 28.9

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

   3. Applied sin-sum 0.8

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

   if -8.131893915803983e-09 < eps < 3.144339914669044e-18

   1. Initial program 44.7

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

   3. Applied diff-sin 44.7

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

   4. Simplified 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.131893915803982637690050267343516710028 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.144339914669044167378723460572493433372 \cdot 10^{-18}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(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)))