cos2 (problem 3.4.1)

Average Error: 31.3 → 0.4

Time: 27.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.03428006622296721206399894299465813674033:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)\right) + \mathsf{fma}\left(\frac{\frac{-1}{x}}{x}, \cos x, \cos x \cdot \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 0.02235098969897789805694188203233352396637:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]

C

double f(double x) {
        double r53794 = 1.0;
        double r53795 = x;
        double r53796 = cos(r53795);
        double r53797 = r53794 - r53796;
        double r53798 = r53795 * r53795;
        double r53799 = r53797 / r53798;
        return r53799;
}

↓

double f(double x) {
        double r53800 = x;
        double r53801 = -0.03428006622296721;
        bool r53802 = r53800 <= r53801;
        double r53803 = 1.0;
        double r53804 = cos(r53800);
        double r53805 = r53803 - r53804;
        double r53806 = r53805 / r53800;
        double r53807 = r53806 / r53800;
        double r53808 = expm1(r53807);
        double r53809 = log1p(r53808);
        double r53810 = -1.0;
        double r53811 = r53810 / r53800;
        double r53812 = r53811 / r53800;
        double r53813 = 1.0;
        double r53814 = r53813 / r53800;
        double r53815 = r53814 / r53800;
        double r53816 = r53804 * r53815;
        double r53817 = fma(r53812, r53804, r53816);
        double r53818 = r53809 + r53817;
        double r53819 = 0.022350989698977898;
        bool r53820 = r53800 <= r53819;
        double r53821 = r53800 * r53800;
        double r53822 = -0.041666666666666664;
        double r53823 = 0.001388888888888889;
        double r53824 = 4.0;
        double r53825 = pow(r53800, r53824);
        double r53826 = 0.5;
        double r53827 = fma(r53823, r53825, r53826);
        double r53828 = fma(r53821, r53822, r53827);
        double r53829 = r53805 / r53821;
        double r53830 = r53820 ? r53828 : r53829;
        double r53831 = r53802 ? r53818 : r53830;
        return r53831;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.03428006622296721

   1. Initial program 1.2

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

   3. Applied div-sub 1.3

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

   4. Simplified 1.4

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

   5. Simplified 0.6

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

   7. Applied *-un-lft-identity 0.6

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

   8. Applied div-inv 0.6

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

   9. Applied times-frac 0.6

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

   10. Applied add-sqr-sqrt 0.7

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

   11. Applied prod-diff 0.6

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{x}}{x}}, \sqrt{\frac{\frac{1}{x}}{x}}, -\frac{\frac{1}{x}}{x} \cdot \frac{\cos x}{1}\right) + \mathsf{fma}\left(-\frac{\frac{1}{x}}{x}, \frac{\cos x}{1}, \frac{\frac{1}{x}}{x} \cdot \frac{\cos x}{1}\right)}\]

   12. Simplified 0.6

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{\frac{1}{x}}{x}}, \sqrt{\frac{\frac{1}{x}}{x}}, \left(-\frac{\frac{1}{x}}{x}\right) \cdot \cos x\right)} + \mathsf{fma}\left(-\frac{\frac{1}{x}}{x}, \frac{\cos x}{1}, \frac{\frac{1}{x}}{x} \cdot \frac{\cos x}{1}\right)\]

   13. Simplified 0.6

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{\frac{1}{x}}{x}}, \sqrt{\frac{\frac{1}{x}}{x}}, \left(-\frac{\frac{1}{x}}{x}\right) \cdot \cos x\right) + \color{blue}{\mathsf{fma}\left(-\frac{\frac{1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right)}\]
   14. Using strategy rm

   15. Applied log1p-expm1-u 0.6

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sqrt{\frac{\frac{1}{x}}{x}}, \sqrt{\frac{\frac{1}{x}}{x}}, \left(-\frac{\frac{1}{x}}{x}\right) \cdot \cos x\right)\right)\right)} + \mathsf{fma}\left(-\frac{\frac{1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right)\]

   16. Simplified 0.5

       \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)}\right) + \mathsf{fma}\left(-\frac{\frac{1}{x}}{x}, \cos x, \frac{\frac{1}{x}}{x} \cdot \cos x\right)\]

   if -0.03428006622296721 < x < 0.022350989698977898

   1. Initial program 62.3

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

   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

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

   if 0.022350989698977898 < x

   1. Initial program 1.0

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.03428006622296721206399894299465813674033:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\frac{1 - \cos x}{x}}{x}\right)\right) + \mathsf{fma}\left(\frac{\frac{-1}{x}}{x}, \cos x, \cos x \cdot \frac{\frac{1}{x}}{x}\right)\\ \mathbf{elif}\;x \le 0.02235098969897789805694188203233352396637:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(\frac{1}{720}, {x}^{4}, \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))