2sin (example 3.3)

Average Error: 37.3 → 0.4

Time: 16.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.038768429931681206984181854539872613685 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 2.920080635356147314623251600384200299398 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r75421 = x;
        double r75422 = eps;
        double r75423 = r75421 + r75422;
        double r75424 = sin(r75423);
        double r75425 = sin(r75421);
        double r75426 = r75424 - r75425;
        return r75426;
}

↓

double f(double x, double eps) {
        double r75427 = eps;
        double r75428 = -1.0387684299316812e-08;
        bool r75429 = r75427 <= r75428;
        double r75430 = 2.9200806353561473e-09;
        bool r75431 = r75427 <= r75430;
        double r75432 = !r75431;
        bool r75433 = r75429 || r75432;
        double r75434 = x;
        double r75435 = sin(r75434);
        double r75436 = cos(r75427);
        double r75437 = r75435 * r75436;
        double r75438 = cos(r75434);
        double r75439 = sin(r75427);
        double r75440 = r75438 * r75439;
        double r75441 = r75437 + r75440;
        double r75442 = r75441 - r75435;
        double r75443 = 2.0;
        double r75444 = r75427 / r75443;
        double r75445 = sin(r75444);
        double r75446 = r75434 + r75427;
        double r75447 = r75446 + r75434;
        double r75448 = r75447 / r75443;
        double r75449 = cos(r75448);
        double r75450 = r75445 * r75449;
        double r75451 = r75443 * r75450;
        double r75452 = r75433 ? r75442 : r75451;
        return r75452;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.3
Target	15.0
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 < -1.0387684299316812e-08 or 2.9200806353561473e-09 < eps

   1. Initial program 30.1

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -1.0387684299316812e-08 < eps < 2.9200806353561473e-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

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.038768429931681206984181854539872613685 \cdot 10^{-8} \lor \neg \left(\varepsilon \le 2.920080635356147314623251600384200299398 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 
(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)))