cos2 (problem 3.4.1)

Average Error: 30.9 → 0.3

Time: 37.8s

Precision: 64

Math

\[\frac{1.0 - \cos x}{x \cdot x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.02712486700916196:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1.0 \cdot \left(1.0 \cdot 1.0\right) - \frac{\cos \left(x + x\right) + 1}{2} \cdot \cos x}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}{x}\\ \mathbf{elif}\;x \le 0.031731941663915054:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{\left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}^{3} - {\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1.0 + \cos x, 1.0 \cdot 1.0\right) \cdot \mathsf{fma}\left(1.0 \cdot \left(1.0 \cdot 1.0\right), 1.0 \cdot \left(1.0 \cdot 1.0\right), \left(\left(\cos x \cdot \cos x\right) \cdot \cos x + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)\right)}}{x}\\ \end{array}\]

C

double f(double x) {
        double r732845 = 1.0;
        double r732846 = x;
        double r732847 = cos(r732846);
        double r732848 = r732845 - r732847;
        double r732849 = r732846 * r732846;
        double r732850 = r732848 / r732849;
        return r732850;
}

↓

double f(double x) {
        double r732851 = x;
        double r732852 = -0.02712486700916196;
        bool r732853 = r732851 <= r732852;
        double r732854 = 1.0;
        double r732855 = r732854 / r732851;
        double r732856 = 1.0;
        double r732857 = r732856 * r732856;
        double r732858 = r732856 * r732857;
        double r732859 = r732851 + r732851;
        double r732860 = cos(r732859);
        double r732861 = r732860 + r732854;
        double r732862 = 2.0;
        double r732863 = r732861 / r732862;
        double r732864 = cos(r732851);
        double r732865 = r732863 * r732864;
        double r732866 = r732858 - r732865;
        double r732867 = r732856 + r732864;
        double r732868 = fma(r732867, r732864, r732857);
        double r732869 = r732866 / r732868;
        double r732870 = r732869 / r732851;
        double r732871 = r732855 * r732870;
        double r732872 = 0.031731941663915054;
        bool r732873 = r732851 <= r732872;
        double r732874 = r732851 * r732851;
        double r732875 = -0.041666666666666664;
        double r732876 = 0.001388888888888889;
        double r732877 = r732876 * r732874;
        double r732878 = 0.5;
        double r732879 = fma(r732874, r732877, r732878);
        double r732880 = fma(r732874, r732875, r732879);
        double r732881 = 3.0;
        double r732882 = pow(r732858, r732881);
        double r732883 = r732864 * r732864;
        double r732884 = r732883 * r732864;
        double r732885 = pow(r732884, r732881);
        double r732886 = r732882 - r732885;
        double r732887 = fma(r732864, r732867, r732857);
        double r732888 = r732884 + r732858;
        double r732889 = r732888 * r732884;
        double r732890 = fma(r732858, r732858, r732889);
        double r732891 = r732887 * r732890;
        double r732892 = r732886 / r732891;
        double r732893 = r732892 / r732851;
        double r732894 = r732855 * r732893;
        double r732895 = r732873 ? r732880 : r732894;
        double r732896 = r732853 ? r732871 : r732895;
        return r732896;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.02712486700916196

   1. Initial program 1.1

      \[\frac{1.0 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left(1.0 - \cos x\right)}}{x \cdot x}\]

   4. Applied times-frac 0.5

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1.0 - \cos x}{x}}\]
   5. Using strategy rm

   6. Applied flip3-- 0.6

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\frac{{1.0}^{3} - {\left(\cos x\right)}^{3}}{1.0 \cdot 1.0 + \left(\cos x \cdot \cos x + 1.0 \cdot \cos x\right)}}}{x}\]

   7. Simplified 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{\color{blue}{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \left(\cos x \cdot \cos x\right)}}{1.0 \cdot 1.0 + \left(\cos x \cdot \cos x + 1.0 \cdot \cos x\right)}}{x}\]

   8. Simplified 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \left(\cos x \cdot \cos x\right)}{\color{blue}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}}{x}\]
   9. Using strategy rm

   10. Applied cos-mult 0.5

       \[\leadsto \frac{1}{x} \cdot \frac{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \color{blue}{\frac{\cos \left(x + x\right) + \cos \left(x - x\right)}{2}}}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}{x}\]

   11. Simplified 0.5

       \[\leadsto \frac{1}{x} \cdot \frac{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \frac{\color{blue}{1 + \cos \left(x + x\right)}}{2}}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}{x}\]

   if -0.02712486700916196 < x < 0.031731941663915054

   1. Initial program 62.2

      \[\frac{1.0 - \cos x}{x \cdot x}\]

   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 0.031731941663915054 < x

   1. Initial program 1.0

      \[\frac{1.0 - \cos x}{x \cdot x}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(1.0 - \cos x\right)}}{x \cdot x}\]

   4. Applied times-frac 0.4

      \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{1.0 - \cos x}{x}}\]
   5. Using strategy rm

   6. Applied flip3-- 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\frac{{1.0}^{3} - {\left(\cos x\right)}^{3}}{1.0 \cdot 1.0 + \left(\cos x \cdot \cos x + 1.0 \cdot \cos x\right)}}}{x}\]

   7. Simplified 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{\color{blue}{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \left(\cos x \cdot \cos x\right)}}{1.0 \cdot 1.0 + \left(\cos x \cdot \cos x + 1.0 \cdot \cos x\right)}}{x}\]

   8. Simplified 0.5

      \[\leadsto \frac{1}{x} \cdot \frac{\frac{\left(1.0 \cdot 1.0\right) \cdot 1.0 - \cos x \cdot \left(\cos x \cdot \cos x\right)}{\color{blue}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}}{x}\]
   9. Using strategy rm

   10. Applied flip3-- 0.5

       \[\leadsto \frac{1}{x} \cdot \frac{\frac{\color{blue}{\frac{{\left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right)}^{3} - {\left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)}^{3}}{\left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) + \left(\left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right) + \left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)\right)}}}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}{x}\]

   11. Applied associate-/l/ 0.5

       \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\frac{{\left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right)}^{3} - {\left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)}^{3}}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right) \cdot \left(\left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) \cdot \left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) + \left(\left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right) + \left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)\right)\right)}}}{x}\]

   12. Simplified 0.5

       \[\leadsto \frac{1}{x} \cdot \frac{\frac{{\left(\left(1.0 \cdot 1.0\right) \cdot 1.0\right)}^{3} - {\left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos x, 1.0 + \cos x, 1.0 \cdot 1.0\right) \cdot \mathsf{fma}\left(1.0 \cdot \left(1.0 \cdot 1.0\right), 1.0 \cdot \left(1.0 \cdot 1.0\right), \left(\cos x \cdot \left(\cos x \cdot \cos x\right)\right) \cdot \left(\cos x \cdot \left(\cos x \cdot \cos x\right) + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right)\right)}}}{x}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.02712486700916196:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1.0 \cdot \left(1.0 \cdot 1.0\right) - \frac{\cos \left(x + x\right) + 1}{2} \cdot \cos x}{\mathsf{fma}\left(1.0 + \cos x, \cos x, 1.0 \cdot 1.0\right)}}{x}\\ \mathbf{elif}\;x \le 0.031731941663915054:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{-1}{24}, \mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \left(x \cdot x\right), \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{{\left(1.0 \cdot \left(1.0 \cdot 1.0\right)\right)}^{3} - {\left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)}^{3}}{\mathsf{fma}\left(\cos x, 1.0 + \cos x, 1.0 \cdot 1.0\right) \cdot \mathsf{fma}\left(1.0 \cdot \left(1.0 \cdot 1.0\right), 1.0 \cdot \left(1.0 \cdot 1.0\right), \left(\left(\cos x \cdot \cos x\right) \cdot \cos x + 1.0 \cdot \left(1.0 \cdot 1.0\right)\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  (/ (- 1.0 (cos x)) (* x x)))