2sin (example 3.3)

Average Error: 37.1 → 0.4

Time: 12.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r120493 = x;
        double r120494 = eps;
        double r120495 = r120493 + r120494;
        double r120496 = sin(r120495);
        double r120497 = sin(r120493);
        double r120498 = r120496 - r120497;
        return r120498;
}

↓

double f(double x, double eps) {
        double r120499 = eps;
        double r120500 = -8.065146023627676e-09;
        bool r120501 = r120499 <= r120500;
        double r120502 = 3.1297217212623504e-09;
        bool r120503 = r120499 <= r120502;
        double r120504 = !r120503;
        bool r120505 = r120501 || r120504;
        double r120506 = x;
        double r120507 = sin(r120506);
        double r120508 = cos(r120499);
        double r120509 = r120507 * r120508;
        double r120510 = cos(r120506);
        double r120511 = sin(r120499);
        double r120512 = r120510 * r120511;
        double r120513 = r120509 + r120512;
        double r120514 = r120513 - r120507;
        double r120515 = 2.0;
        double r120516 = r120499 / r120515;
        double r120517 = sin(r120516);
        double r120518 = r120506 + r120499;
        double r120519 = r120518 + r120506;
        double r120520 = r120519 / r120515;
        double r120521 = cos(r120520);
        double r120522 = r120517 * r120521;
        double r120523 = r120515 * r120522;
        double r120524 = r120505 ? r120514 : r120523;
        return r120524;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
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 < -8.065146023627676e-09 or 3.1297217212623504e-09 < eps

   1. Initial program 29.9

      \[\sin \left(x + \varepsilon\right) - \sin x\]
   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 -8.065146023627676e-09 < eps < 3.1297217212623504e-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

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -8.0651460236276763 \cdot 10^{-9} \lor \neg \left(\varepsilon \le 3.1297217212623504 \cdot 10^{-9}\right):\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(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)))