cos2 (problem 3.4.1)

Average Error: 31.4 → 0.3

Time: 30.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02740531929745840256096300890931161120534:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \mathbf{elif}\;x \le 0.03110675776409119533405522872726578498259:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{1 - \cos x}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]

C

double f(double x) {
        double r762331 = 1.0;
        double r762332 = x;
        double r762333 = cos(r762332);
        double r762334 = r762331 - r762333;
        double r762335 = r762332 * r762332;
        double r762336 = r762334 / r762335;
        return r762336;
}

↓

double f(double x) {
        double r762337 = x;
        double r762338 = -0.027405319297458403;
        bool r762339 = r762337 <= r762338;
        double r762340 = 1.0;
        double r762341 = cos(r762337);
        double r762342 = r762340 - r762341;
        double r762343 = exp(r762342);
        double r762344 = log(r762343);
        double r762345 = sqrt(r762344);
        double r762346 = r762345 / r762337;
        double r762347 = r762346 / r762337;
        double r762348 = sqrt(r762342);
        double r762349 = r762347 * r762348;
        double r762350 = 0.031106757764091195;
        bool r762351 = r762337 <= r762350;
        double r762352 = r762337 * r762337;
        double r762353 = -0.041666666666666664;
        double r762354 = 0.001388888888888889;
        double r762355 = r762352 * r762354;
        double r762356 = 0.5;
        double r762357 = fma(r762352, r762355, r762356);
        double r762358 = fma(r762352, r762353, r762357);
        double r762359 = r762348 / r762337;
        double r762360 = r762359 / r762337;
        double r762361 = r762360 * r762348;
        double r762362 = r762351 ? r762358 : r762361;
        double r762363 = r762339 ? r762349 : r762362;
        return r762363;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.027405319297458403

   1. Initial program 1.2

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

   3. Applied associate-/r* 0.5

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

   5. Applied *-un-lft-identity 0.5

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

   6. Applied *-un-lft-identity 0.5

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

   7. Applied add-sqr-sqrt 0.6

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

   8. Applied times-frac 0.6

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

   9. Applied times-frac 0.6

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

   10. Simplified 0.6

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

   12. Applied add-log-exp 0.6

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

   13. Applied add-log-exp 0.6

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

   14. Applied diff-log 0.6

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

   15. Simplified 0.6

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

   if -0.027405319297458403 < x < 0.031106757764091195

   1. Initial program 62.4

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

   2. Taylor expanded around 0 0.0

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

   3. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)}\]

   if 0.031106757764091195 < x

   1. Initial program 1.0

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

   3. Applied associate-/r* 0.5

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

   5. Applied *-un-lft-identity 0.5

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

   6. Applied *-un-lft-identity 0.5

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

   7. Applied add-sqr-sqrt 0.6

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

   8. Applied times-frac 0.6

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

   9. Applied times-frac 0.6

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

   10. Simplified 0.6

       \[\leadsto \color{blue}{\sqrt{1 - \cos x}} \cdot \frac{\frac{\sqrt{1 - \cos x}}{x}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02740531929745840256096300890931161120534:\\ \;\;\;\;\frac{\frac{\sqrt{\log \left(e^{1 - \cos x}\right)}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \mathbf{elif}\;x \le 0.03110675776409119533405522872726578498259:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{1}{720}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{1 - \cos x}}{x}}{x} \cdot \sqrt{1 - \cos x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))