2cos (problem 3.3.5)

Average Error: 39.2 → 0.8

Time: 20.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001322493479089982:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00010993973433856837:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

C

double f(double x, double eps) {
        double r991320 = x;
        double r991321 = eps;
        double r991322 = r991320 + r991321;
        double r991323 = cos(r991322);
        double r991324 = cos(r991320);
        double r991325 = r991323 - r991324;
        return r991325;
}

↓

double f(double x, double eps) {
        double r991326 = eps;
        double r991327 = -0.0001322493479089982;
        bool r991328 = r991326 <= r991327;
        double r991329 = x;
        double r991330 = cos(r991329);
        double r991331 = cos(r991326);
        double r991332 = r991330 * r991331;
        double r991333 = sin(r991329);
        double r991334 = sin(r991326);
        double r991335 = r991333 * r991334;
        double r991336 = r991332 - r991335;
        double r991337 = r991336 - r991330;
        double r991338 = 0.00010993973433856837;
        bool r991339 = r991326 <= r991338;
        double r991340 = 2.0;
        double r991341 = r991326 / r991340;
        double r991342 = sin(r991341);
        double r991343 = r991329 + r991326;
        double r991344 = r991343 + r991329;
        double r991345 = r991344 / r991340;
        double r991346 = sin(r991345);
        double r991347 = log1p(r991346);
        double r991348 = expm1(r991347);
        double r991349 = r991342 * r991348;
        double r991350 = -2.0;
        double r991351 = r991349 * r991350;
        double r991352 = r991339 ? r991351 : r991337;
        double r991353 = r991328 ? r991337 : r991352;
        return r991353;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -0.0001322493479089982 or 0.00010993973433856837 < eps

   1. Initial program 30.1

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

   3. Applied cos-sum 0.9

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

   if -0.0001322493479089982 < eps < 0.00010993973433856837

   1. Initial program 48.9

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

   3. Applied diff-cos 37.6

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

   4. Simplified 0.6

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

   6. Applied expm1-log1p-u 0.6

      \[\leadsto -2 \cdot \left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{x + \left(\varepsilon + x\right)}{2}\right)\right)\right)\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -0.0001322493479089982:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \le 0.00010993973433856837:\\ \;\;\;\;\left(\sin \left(\frac{\varepsilon}{2}\right) \cdot \mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left(\sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)\right)\right)\right)\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  (- (cos (+ x eps)) (cos x)))