2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 20.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r54404 = x;
        double r54405 = eps;
        double r54406 = r54404 + r54405;
        double r54407 = sin(r54406);
        double r54408 = sin(r54404);
        double r54409 = r54407 - r54408;
        return r54409;
}

↓

double f(double x, double eps) {
        double r54410 = eps;
        double r54411 = -0.00815188689103541;
        bool r54412 = r54410 <= r54411;
        double r54413 = 8.837665134522453e-09;
        bool r54414 = r54410 <= r54413;
        double r54415 = !r54414;
        bool r54416 = r54412 || r54415;
        double r54417 = x;
        double r54418 = sin(r54417);
        double r54419 = cos(r54410);
        double r54420 = r54418 * r54419;
        double r54421 = cos(r54417);
        double r54422 = sin(r54410);
        double r54423 = r54421 * r54422;
        double r54424 = r54420 + r54423;
        double r54425 = r54424 - r54418;
        double r54426 = 2.0;
        double r54427 = r54410 / r54426;
        double r54428 = sin(r54427);
        double r54429 = r54417 + r54410;
        double r54430 = r54429 + r54417;
        double r54431 = r54430 / r54426;
        double r54432 = cos(r54431);
        double r54433 = r54428 * r54432;
        double r54434 = r54426 * r54433;
        double r54435 = r54416 ? r54425 : r54434;
        return r54435;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.2
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 < -0.00815188689103541 or 8.837665134522453e-09 < eps

   1. Initial program 30.3

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

   3. Applied sin-sum 0.4

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

   if -0.00815188689103541 < eps < 8.837665134522453e-09

   1. Initial program 44.1

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

   3. Applied diff-sin 44.1

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

   4. Simplified 0.4

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.008151886891035409199446348793571814894676 \lor \neg \left(\varepsilon \le 8.837665134522452590648921417169903147482 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))