2sin (example 3.3)

Average Error: 37.1 → 0.8

Time: 22.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -40694913.773095824:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.386517640886532 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r3634525 = x;
        double r3634526 = eps;
        double r3634527 = r3634525 + r3634526;
        double r3634528 = sin(r3634527);
        double r3634529 = sin(r3634525);
        double r3634530 = r3634528 - r3634529;
        return r3634530;
}

↓

double f(double x, double eps) {
        double r3634531 = eps;
        double r3634532 = -40694913.773095824;
        bool r3634533 = r3634531 <= r3634532;
        double r3634534 = x;
        double r3634535 = sin(r3634534);
        double r3634536 = cos(r3634531);
        double r3634537 = r3634535 * r3634536;
        double r3634538 = cos(r3634534);
        double r3634539 = sin(r3634531);
        double r3634540 = r3634538 * r3634539;
        double r3634541 = r3634537 + r3634540;
        double r3634542 = r3634541 - r3634535;
        double r3634543 = 6.386517640886532e-16;
        bool r3634544 = r3634531 <= r3634543;
        double r3634545 = 2.0;
        double r3634546 = r3634531 / r3634545;
        double r3634547 = sin(r3634546);
        double r3634548 = r3634534 + r3634531;
        double r3634549 = r3634534 + r3634548;
        double r3634550 = r3634549 / r3634545;
        double r3634551 = cos(r3634550);
        double r3634552 = r3634547 * r3634551;
        double r3634553 = r3634545 * r3634552;
        double r3634554 = r3634544 ? r3634553 : r3634542;
        double r3634555 = r3634533 ? r3634542 : r3634554;
        return r3634555;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.3
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if eps < -40694913.773095824 or 6.386517640886532e-16 < eps

   1. Initial program 29.7

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -40694913.773095824 < eps < 6.386517640886532e-16

   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 0.9

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -40694913.773095824:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \le 6.386517640886532 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(FPCore (x eps)
  :name "2sin (example 3.3)"

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

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