cos2 (problem 3.4.1)

Average Error: 31.6 → 0.3

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\sqrt[3]{{\left(\cos x \cdot \left(\cos x + 1\right)\right)}^{3}} + 1 \cdot 1\right) \cdot x}\\ \mathbf{elif}\;x \le 0.0284410596258715502:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x}\\ \end{array}\]

C

double f(double x) {
        double r17089 = 1.0;
        double r17090 = x;
        double r17091 = cos(r17090);
        double r17092 = r17089 - r17091;
        double r17093 = r17090 * r17090;
        double r17094 = r17092 / r17093;
        return r17094;
}

↓

double f(double x) {
        double r17095 = x;
        double r17096 = -0.02660967226737158;
        bool r17097 = r17095 <= r17096;
        double r17098 = 1.0;
        double r17099 = r17098 / r17095;
        double r17100 = 1.0;
        double r17101 = 3.0;
        double r17102 = pow(r17100, r17101);
        double r17103 = cos(r17095);
        double r17104 = pow(r17103, r17101);
        double r17105 = r17102 - r17104;
        double r17106 = r17103 + r17100;
        double r17107 = r17103 * r17106;
        double r17108 = pow(r17107, r17101);
        double r17109 = cbrt(r17108);
        double r17110 = r17100 * r17100;
        double r17111 = r17109 + r17110;
        double r17112 = r17111 * r17095;
        double r17113 = r17105 / r17112;
        double r17114 = r17099 * r17113;
        double r17115 = 0.02844105962587155;
        bool r17116 = r17095 <= r17115;
        double r17117 = 0.001388888888888889;
        double r17118 = 4.0;
        double r17119 = pow(r17095, r17118);
        double r17120 = r17117 * r17119;
        double r17121 = 0.5;
        double r17122 = r17120 + r17121;
        double r17123 = 0.041666666666666664;
        double r17124 = 2.0;
        double r17125 = pow(r17095, r17124);
        double r17126 = r17123 * r17125;
        double r17127 = r17122 - r17126;
        double r17128 = exp(r17105);
        double r17129 = log(r17128);
        double r17130 = r17107 + r17110;
        double r17131 = r17130 * r17095;
        double r17132 = r17129 / r17131;
        double r17133 = r17099 * r17132;
        double r17134 = r17116 ? r17127 : r17133;
        double r17135 = r17097 ? r17114 : r17134;
        return r17135;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.02660967226737158

   1. Initial program 0.9

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

   3. Applied *-un-lft-identity 0.9

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

   4. Applied times-frac 0.5

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

   6. Applied flip3-- 0.5

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

   7. Applied associate-/l/ 0.5

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

   8. Simplified 0.5

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

   10. Applied add-cbrt-cube 0.5

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

   11. Applied add-cbrt-cube 0.5

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

   12. Applied cbrt-unprod 0.5

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

   13. Simplified 0.5

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

   if -0.02660967226737158 < x < 0.02844105962587155

   1. Initial program 62.2

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

   2. Taylor expanded around 0 0.0

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

   if 0.02844105962587155 < 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 flip3-- 0.5

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

   7. Applied associate-/l/ 0.5

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

   8. Simplified 0.5

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

   10. Applied add-log-exp 0.5

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

   11. Applied add-log-exp 0.5

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

   12. Applied diff-log 0.6

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

   13. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0266096722673715806:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{1}^{3} - {\left(\cos x\right)}^{3}}{\left(\sqrt[3]{{\left(\cos x \cdot \left(\cos x + 1\right)\right)}^{3}} + 1 \cdot 1\right) \cdot x}\\ \mathbf{elif}\;x \le 0.0284410596258715502:\\ \;\;\;\;\left(\frac{1}{720} \cdot {x}^{4} + \frac{1}{2}\right) - \frac{1}{24} \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\log \left(e^{{1}^{3} - {\left(\cos x\right)}^{3}}\right)}{\left(\cos x \cdot \left(\cos x + 1\right) + 1 \cdot 1\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1 (cos x)) (* x x)))