2cos (problem 3.3.5)

Average Error: 40.0 → 16.4

Time: 7.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.153154815626698153605163400370998364602 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 1.280988864612255259754335784405654408147 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \end{array}\]

C

double f(double x, double eps) {
        double r50345 = x;
        double r50346 = eps;
        double r50347 = r50345 + r50346;
        double r50348 = cos(r50347);
        double r50349 = cos(r50345);
        double r50350 = r50348 - r50349;
        return r50350;
}

↓

double f(double x, double eps) {
        double r50351 = eps;
        double r50352 = -1.1531548156266982e-10;
        bool r50353 = r50351 <= r50352;
        double r50354 = 1.2809888646122553e-06;
        bool r50355 = r50351 <= r50354;
        double r50356 = !r50355;
        bool r50357 = r50353 || r50356;
        double r50358 = x;
        double r50359 = cos(r50358);
        double r50360 = cos(r50351);
        double r50361 = r50359 * r50360;
        double r50362 = expm1(r50361);
        double r50363 = log1p(r50362);
        double r50364 = sin(r50358);
        double r50365 = sin(r50351);
        double r50366 = r50364 * r50365;
        double r50367 = r50363 - r50366;
        double r50368 = r50367 - r50359;
        double r50369 = 0.041666666666666664;
        double r50370 = 4.0;
        double r50371 = pow(r50351, r50370);
        double r50372 = 0.5;
        double r50373 = 2.0;
        double r50374 = pow(r50351, r50373);
        double r50375 = r50372 * r50374;
        double r50376 = fma(r50358, r50351, r50375);
        double r50377 = -r50376;
        double r50378 = fma(r50369, r50371, r50377);
        double r50379 = r50361 - r50366;
        double r50380 = r50379 + r50359;
        double r50381 = r50378 * r50380;
        double r50382 = r50381 / r50380;
        double r50383 = r50357 ? r50368 : r50382;
        return r50383;
}

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if eps < -1.1531548156266982e-10 or 1.2809888646122553e-06 < eps

   1. Initial program 30.5

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

   3. Applied cos-sum 1.2

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied log1p-expm1-u 1.3

      \[\leadsto \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right)} - \sin x \cdot \sin \varepsilon\right) - \cos x\]

   if -1.1531548156266982e-10 < eps < 1.2809888646122553e-06

   1. Initial program 49.7

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

   3. Applied cos-sum 49.3

      \[\leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   4. Using strategy rm

   5. Applied flip-- 49.3

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

   6. Simplified 49.3

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

   7. Taylor expanded around 0 32.0

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {\varepsilon}^{4} - \left(x \cdot \varepsilon + \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\]

   8. Simplified 32.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)} \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.153154815626698153605163400370998364602 \cdot 10^{-10} \lor \neg \left(\varepsilon \le 1.280988864612255259754335784405654408147 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x \cdot \cos \varepsilon\right)\right) - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{24}, {\varepsilon}^{4}, -\mathsf{fma}\left(x, \varepsilon, \frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot \left(\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x\right)}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) + \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))