2cos (problem 3.3.5)

Average Error: 39.5 → 0.7

Time: 15.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00391800888554098:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

C

double f(double x, double eps) {
        double r742391 = x;
        double r742392 = eps;
        double r742393 = r742391 + r742392;
        double r742394 = cos(r742393);
        double r742395 = cos(r742391);
        double r742396 = r742394 - r742395;
        return r742396;
}

↓

double f(double x, double eps) {
        double r742397 = eps;
        double r742398 = -0.22306452923435388;
        bool r742399 = r742397 <= r742398;
        double r742400 = x;
        double r742401 = cos(r742400);
        double r742402 = cos(r742397);
        double r742403 = r742401 * r742402;
        double r742404 = sin(r742400);
        double r742405 = sin(r742397);
        double r742406 = r742404 * r742405;
        double r742407 = r742401 + r742406;
        double r742408 = r742403 - r742407;
        double r742409 = 0.00391800888554098;
        bool r742410 = r742397 <= r742409;
        double r742411 = r742400 + r742397;
        double r742412 = r742400 + r742411;
        double r742413 = 2.0;
        double r742414 = r742412 / r742413;
        double r742415 = sin(r742414);
        double r742416 = -2.0;
        double r742417 = r742397 / r742413;
        double r742418 = sin(r742417);
        double r742419 = r742416 * r742418;
        double r742420 = r742415 * r742419;
        double r742421 = r742410 ? r742420 : r742408;
        double r742422 = r742399 ? r742408 : r742421;
        return r742422;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -0.22306452923435388 or 0.00391800888554098 < eps

   1. Initial program 29.7

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

   3. Applied cos-sum 0.8

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

   4. Applied associate--l- 0.8

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

   if -0.22306452923435388 < eps < 0.00391800888554098

   1. Initial program 49.0

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

   3. Applied diff-cos 37.2

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

   4. Simplified 0.7

      \[\leadsto -2 \cdot \color{blue}{\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right)\right)}\]
   5. Using strategy rm

   6. Applied associate-*r* 0.6

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.22306452923435388:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \le 0.00391800888554098:\\ \;\;\;\;\sin \left(\frac{x + \left(x + \varepsilon\right)}{2}\right) \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))