2nthrt (problem 3.4.6)

Average Error: 29.4 → 21.9

Time: 15.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 4.773292077298795e+235
  2. n = 2.4612524637435313e-181

Math

\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r53786 = x;
        double r53787 = 1.0;
        double r53788 = r53786 + r53787;
        double r53789 = n;
        double r53790 = r53787 / r53789;
        double r53791 = pow(r53788, r53790);
        double r53792 = pow(r53786, r53790);
        double r53793 = r53791 - r53792;
        return r53793;
}

↓

double f(double x, double n) {
        double r53794 = 1.0;
        double r53795 = n;
        double r53796 = r53794 / r53795;
        double r53797 = -2.429908229671821e-07;
        bool r53798 = r53796 <= r53797;
        double r53799 = 1.4807923677373035e-15;
        bool r53800 = r53796 <= r53799;
        double r53801 = !r53800;
        bool r53802 = r53798 || r53801;
        double r53803 = x;
        double r53804 = r53803 + r53794;
        double r53805 = pow(r53804, r53796);
        double r53806 = pow(r53803, r53796);
        double r53807 = r53805 - r53806;
        double r53808 = 3.0;
        double r53809 = pow(r53807, r53808);
        double r53810 = cbrt(r53809);
        double r53811 = pow(r53810, r53808);
        double r53812 = cbrt(r53811);
        double r53813 = pow(r53812, r53808);
        double r53814 = cbrt(r53813);
        double r53815 = pow(r53814, r53808);
        double r53816 = cbrt(r53815);
        double r53817 = r53796 / r53803;
        double r53818 = log(r53803);
        double r53819 = -r53818;
        double r53820 = 2.0;
        double r53821 = pow(r53795, r53820);
        double r53822 = r53803 * r53821;
        double r53823 = r53819 / r53822;
        double r53824 = 0.5;
        double r53825 = pow(r53803, r53820);
        double r53826 = r53825 * r53795;
        double r53827 = r53824 / r53826;
        double r53828 = fma(r53823, r53794, r53827);
        double r53829 = r53817 - r53828;
        double r53830 = r53802 ? r53816 : r53829;
        return r53830;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 n) < -2.429908229671821e-07 or 1.4807923677373035e-15 < (/ 1.0 n)

   1. Initial program 9.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 9.4

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

   4. Simplified 9.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 9.4

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

   7. Simplified 9.4

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\right)}^{3}}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 9.4

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

   10. Simplified 9.4

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}}\right)}^{3}}\right)}^{3}}\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 9.4

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

   13. Simplified 9.4

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

   if -2.429908229671821e-07 < (/ 1.0 n) < 1.4807923677373035e-15

   1. Initial program 44.9

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube 44.9

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

   4. Simplified 44.9

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

   5. Taylor expanded around inf 32.2

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

   6. Simplified 31.6

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))