cos2 (problem 3.4.1)

Average Error: 31.9 → 0.3

Time: 20.9s

Precision: 64

Math

\[\frac{1 - \cos x}{x \cdot x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

C

double f(double x) {
        double r868391 = 1.0;
        double r868392 = x;
        double r868393 = cos(r868392);
        double r868394 = r868391 - r868393;
        double r868395 = r868392 * r868392;
        double r868396 = r868394 / r868395;
        return r868396;
}

↓

double f(double x) {
        double r868397 = x;
        double r868398 = -0.03115379972147734;
        bool r868399 = r868397 <= r868398;
        double r868400 = 1.0;
        double r868401 = cos(r868397);
        double r868402 = r868400 - r868401;
        double r868403 = r868402 / r868397;
        double r868404 = r868403 / r868397;
        double r868405 = 0.027390056951785625;
        bool r868406 = r868397 <= r868405;
        double r868407 = -0.041666666666666664;
        double r868408 = r868397 * r868397;
        double r868409 = r868408 * r868408;
        double r868410 = 0.001388888888888889;
        double r868411 = 0.5;
        double r868412 = fma(r868409, r868410, r868411);
        double r868413 = fma(r868407, r868408, r868412);
        double r868414 = r868406 ? r868413 : r868404;
        double r868415 = r868399 ? r868404 : r868414;
        return r868415;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -0.03115379972147734 or 0.027390056951785625 < x

   1. Initial program 1.0

      \[\frac{1 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]

   4. Applied times-frac 0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]
   5. Using strategy rm

   6. Applied associate-*r/ 0.5

      \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}}\]

   7. Simplified 0.5

      \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{x}}}{x}\]

   if -0.03115379972147734 < x < 0.027390056951785625

   1. Initial program 62.4

      \[\frac{1 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 62.4

      \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - \cos x\right)}}{x \cdot x}\]

   4. Applied times-frac 61.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1 - \cos x}{x}}\]

   5. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}}\]

   6. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03115379972147733905751820771001803223044:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \le 0.02739005695178562543867784029316680971533:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{24}, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))