2sin (example 3.3)

Average Error: 37.0 → 0.5

Time: 16.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

C

double f(double x, double eps) {
        double r67546 = x;
        double r67547 = eps;
        double r67548 = r67546 + r67547;
        double r67549 = sin(r67548);
        double r67550 = sin(r67546);
        double r67551 = r67549 - r67550;
        return r67551;
}

↓

double f(double x, double eps) {
        double r67552 = eps;
        double r67553 = -6.6823151206119754e-09;
        bool r67554 = r67552 <= r67553;
        double r67555 = x;
        double r67556 = sin(r67555);
        double r67557 = cos(r67552);
        double r67558 = r67556 * r67557;
        double r67559 = cos(r67555);
        double r67560 = sin(r67552);
        double r67561 = r67559 * r67560;
        double r67562 = r67561 - r67556;
        double r67563 = r67558 + r67562;
        double r67564 = 7.436477518387171e-09;
        bool r67565 = r67552 <= r67564;
        double r67566 = 2.0;
        double r67567 = r67552 / r67566;
        double r67568 = sin(r67567);
        double r67569 = r67555 + r67552;
        double r67570 = r67569 + r67555;
        double r67571 = r67570 / r67566;
        double r67572 = cos(r67571);
        double r67573 = r67568 * r67572;
        double r67574 = r67573 * r67566;
        double r67575 = r67558 + r67561;
        double r67576 = r67575 - r67556;
        double r67577 = r67565 ? r67574 : r67576;
        double r67578 = r67554 ? r67563 : r67577;
        return r67578;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.0
Target	15.1
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 3 regimes

2. if eps < -6.6823151206119754e-09

   1. Initial program 30.1

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

   3. Applied sin-sum 0.6

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

   4. Applied associate--l+ 0.7

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

   if -6.6823151206119754e-09 < eps < 7.436477518387171e-09

   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

      \[\leadsto 2 \cdot \color{blue}{\left(\cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right) \cdot \sin \left(\frac{0 + \varepsilon}{2}\right)\right)}\]

   if 7.436477518387171e-09 < eps

   1. Initial program 29.2

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

   3. Applied sin-sum 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -6.6823151206119754 \cdot 10^{-9}:\\ \;\;\;\;\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)\\ \mathbf{elif}\;\varepsilon \le 7.4364775183871708 \cdot 10^{-9}:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))