2sin (example 3.3)

Average Error: 37.1 → 0.5

Time: 19.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -3.523695819775611137291554397776849327784 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.06104011737916238035658440629202189931 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r94435 = x;
        double r94436 = eps;
        double r94437 = r94435 + r94436;
        double r94438 = sin(r94437);
        double r94439 = sin(r94435);
        double r94440 = r94438 - r94439;
        return r94440;
}

↓

double f(double x, double eps) {
        double r94441 = eps;
        double r94442 = -3.523695819775611e-06;
        bool r94443 = r94441 <= r94442;
        double r94444 = 4.0610401173791624e-11;
        bool r94445 = r94441 <= r94444;
        double r94446 = !r94445;
        bool r94447 = r94443 || r94446;
        double r94448 = x;
        double r94449 = sin(r94448);
        double r94450 = cos(r94441);
        double r94451 = r94449 * r94450;
        double r94452 = cos(r94448);
        double r94453 = sin(r94441);
        double r94454 = r94452 * r94453;
        double r94455 = r94451 + r94454;
        double r94456 = r94455 - r94449;
        double r94457 = 2.0;
        double r94458 = r94441 / r94457;
        double r94459 = sin(r94458);
        double r94460 = r94448 + r94441;
        double r94461 = r94460 + r94448;
        double r94462 = r94461 / r94457;
        double r94463 = cos(r94462);
        double r94464 = r94459 * r94463;
        double r94465 = r94457 * r94464;
        double r94466 = r94447 ? r94456 : r94465;
        return r94466;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	14.9
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 < -3.523695819775611e-06 or 4.0610401173791624e-11 < eps

   1. Initial program 29.6

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -3.523695819775611e-06 < eps < 4.0610401173791624e-11

   1. Initial program 45.0

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

   3. Applied diff-sin 45.0

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

   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 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -3.523695819775611137291554397776849327784 \cdot 10^{-6} \lor \neg \left(\varepsilon \le 4.06104011737916238035658440629202189931 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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)))