2nthrt (problem 3.4.6)

Average Error: 29.5 → 23.0

Time: 20.4s

Precision: 64

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

• could not determine a ground truth for program body

  1. x = 1.61157953583811e+280
  2. n = 1.4915374553808526e-197

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -4392111429010646530 \lor \neg \left(n \le 142153.58580214932\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r62103 = x;
        double r62104 = 1.0;
        double r62105 = r62103 + r62104;
        double r62106 = n;
        double r62107 = r62104 / r62106;
        double r62108 = pow(r62105, r62107);
        double r62109 = pow(r62103, r62107);
        double r62110 = r62108 - r62109;
        return r62110;
}

↓

double f(double x, double n) {
        double r62111 = n;
        double r62112 = -4.3921114290106465e+18;
        bool r62113 = r62111 <= r62112;
        double r62114 = 142153.58580214932;
        bool r62115 = r62111 <= r62114;
        double r62116 = !r62115;
        bool r62117 = r62113 || r62116;
        double r62118 = 1.0;
        double r62119 = x;
        double r62120 = r62118 / r62119;
        double r62121 = 1.0;
        double r62122 = r62121 / r62111;
        double r62123 = log(r62119);
        double r62124 = -r62123;
        double r62125 = 2.0;
        double r62126 = pow(r62111, r62125);
        double r62127 = r62124 / r62126;
        double r62128 = r62122 - r62127;
        double r62129 = r62120 * r62128;
        double r62130 = 0.5;
        double r62131 = pow(r62119, r62125);
        double r62132 = r62131 * r62111;
        double r62133 = r62130 / r62132;
        double r62134 = r62129 - r62133;
        double r62135 = r62119 + r62118;
        double r62136 = r62118 / r62111;
        double r62137 = pow(r62135, r62136);
        double r62138 = pow(r62119, r62136);
        double r62139 = r62137 - r62138;
        double r62140 = exp(r62139);
        double r62141 = cbrt(r62140);
        double r62142 = log(r62141);
        double r62143 = r62125 * r62142;
        double r62144 = 0.3333333333333333;
        double r62145 = log(r62140);
        double r62146 = r62144 * r62145;
        double r62147 = r62143 + r62146;
        double r62148 = r62117 ? r62134 : r62147;
        return r62148;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -4.3921114290106465e+18 or 142153.58580214932 < n

   1. Initial program 44.7

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

   2. Taylor expanded around inf 33.3

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

   3. Simplified 32.9

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

   if -4.3921114290106465e+18 < n < 142153.58580214932

   1. Initial program 9.9

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

   3. Applied add-log-exp 10.0

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

   4. Applied add-log-exp 10.0

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

   5. Applied diff-log 10.1

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

   6. Simplified 10.0

      \[\leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 10.4

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

   9. Applied log-prod 10.4

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

   10. Simplified 10.4

       \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)} + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\]
   11. Using strategy rm

   12. Applied pow1/3 10.3

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \color{blue}{\left({\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{\frac{1}{3}}\right)}\]

   13. Applied log-pow 10.3

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \color{blue}{\frac{1}{3} \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]

   14. Simplified 10.3

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \color{blue}{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}\]
   15. Using strategy rm

   16. Applied add-log-exp 10.4

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\right)\]

   17. Applied add-log-exp 10.3

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \left(\color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\right)\]

   18. Applied diff-log 10.3

       \[\leadsto 2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

   19. Simplified 10.3

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

4. Final simplification 23.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -4392111429010646530 \lor \neg \left(n \le 142153.58580214932\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \frac{1}{3} \cdot \log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))