cos2 (problem 3.4.1)

Average Error: 31.3 → 0.3

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.0300611177250478985:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{1}{\frac{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}{1}}\right)\\ \end{array}\]

C

double f(double x) {
        double r40816 = 1.0;
        double r40817 = x;
        double r40818 = cos(r40817);
        double r40819 = r40816 - r40818;
        double r40820 = r40817 * r40817;
        double r40821 = r40819 / r40820;
        return r40821;
}

↓

double f(double x) {
        double r40822 = x;
        double r40823 = -0.03128865823944801;
        bool r40824 = r40822 <= r40823;
        double r40825 = 1.0;
        double r40826 = r40825 / r40822;
        double r40827 = 1.0;
        double r40828 = cos(r40822);
        double r40829 = r40827 - r40828;
        double r40830 = r40822 / r40829;
        double r40831 = r40825 / r40830;
        double r40832 = r40826 * r40831;
        double r40833 = 0.0300611177250479;
        bool r40834 = r40822 <= r40833;
        double r40835 = 4.0;
        double r40836 = pow(r40822, r40835);
        double r40837 = 0.001388888888888889;
        double r40838 = 0.5;
        double r40839 = 0.041666666666666664;
        double r40840 = 2.0;
        double r40841 = pow(r40822, r40840);
        double r40842 = r40839 * r40841;
        double r40843 = r40838 - r40842;
        double r40844 = fma(r40836, r40837, r40843);
        double r40845 = 3.0;
        double r40846 = pow(r40827, r40845);
        double r40847 = pow(r40828, r40845);
        double r40848 = r40846 - r40847;
        double r40849 = r40822 / r40848;
        double r40850 = r40825 / r40849;
        double r40851 = r40827 + r40828;
        double r40852 = r40827 * r40827;
        double r40853 = fma(r40828, r40851, r40852);
        double r40854 = r40853 / r40825;
        double r40855 = r40825 / r40854;
        double r40856 = r40850 * r40855;
        double r40857 = r40826 * r40856;
        double r40858 = r40834 ? r40844 : r40857;
        double r40859 = r40824 ? r40832 : r40858;
        return r40859;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.03128865823944801

   1. Initial program 0.8

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

   3. Applied *-un-lft-identity 0.8

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

   4. Applied times-frac 0.5

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

   6. Applied clear-num 0.5

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

   if -0.03128865823944801 < x < 0.0300611177250479

   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

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

   if 0.0300611177250479 < x

   1. Initial program 1.1

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

   3. Applied *-un-lft-identity 1.1

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

   4. Applied times-frac 0.6

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

   6. Applied clear-num 0.6

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

   8. Applied flip3-- 0.6

      \[\leadsto \frac{1}{x} \cdot \frac{1}{\frac{x}{\color{blue}{\frac{{1}^{3} - {\left(\cos x\right)}^{3}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}}}}\]

   9. Applied associate-/r/ 0.6

      \[\leadsto \frac{1}{x} \cdot \frac{1}{\color{blue}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}}\]

   10. Applied add-sqr-sqrt 0.6

       \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}} \cdot \left(1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)\right)}\]

   11. Applied times-frac 0.6

       \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\frac{\sqrt{1}}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{\sqrt{1}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\right)}\]

   12. Simplified 0.6

       \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\frac{1}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}}} \cdot \frac{\sqrt{1}}{1 \cdot 1 + \left(\cos x \cdot \cos x + 1 \cdot \cos x\right)}\right)\]

   13. Simplified 0.6

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.031288658239448007:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{\frac{x}{1 - \cos x}}\\ \mathbf{elif}\;x \le 0.0300611177250478985:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, \frac{1}{720}, \frac{1}{2} - \frac{1}{24} \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{\frac{x}{{1}^{3} - {\left(\cos x\right)}^{3}}} \cdot \frac{1}{\frac{\mathsf{fma}\left(\cos x, 1 + \cos x, 1 \cdot 1\right)}{1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))