2sin (example 3.3)

Average Error: 36.9 → 0.6

Time: 21.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double eps) {
        double r3860290 = x;
        double r3860291 = eps;
        double r3860292 = r3860290 + r3860291;
        double r3860293 = sin(r3860292);
        double r3860294 = sin(r3860290);
        double r3860295 = r3860293 - r3860294;
        return r3860295;
}

↓

double f(double x, double eps) {
        double r3860296 = eps;
        double r3860297 = -5.893930927518443e-09;
        bool r3860298 = r3860296 <= r3860297;
        double r3860299 = x;
        double r3860300 = sin(r3860299);
        double r3860301 = cos(r3860296);
        double r3860302 = r3860300 * r3860301;
        double r3860303 = cos(r3860299);
        double r3860304 = sin(r3860296);
        double r3860305 = r3860303 * r3860304;
        double r3860306 = r3860302 + r3860305;
        double r3860307 = r3860306 - r3860300;
        double r3860308 = 1.8942327691411048e-20;
        bool r3860309 = r3860296 <= r3860308;
        double r3860310 = 2.0;
        double r3860311 = r3860299 + r3860296;
        double r3860312 = r3860311 + r3860299;
        double r3860313 = r3860312 / r3860310;
        double r3860314 = cos(r3860313);
        double r3860315 = 0.5;
        double r3860316 = r3860296 * r3860315;
        double r3860317 = sin(r3860316);
        double r3860318 = r3860314 * r3860317;
        double r3860319 = r3860310 * r3860318;
        double r3860320 = r3860309 ? r3860319 : r3860307;
        double r3860321 = r3860298 ? r3860307 : r3860320;
        return r3860321;
}

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

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

4. Final simplification 0.6

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

Reproduce

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

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

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