2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.2

Time: 29.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.6185456715928e+208
  2. n = 6.482235930800528e-56

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -3406.416978222420766542199999094009399414 \lor \neg \left(n \le 1031392298459.7406005859375\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double n) {
        double r61050 = x;
        double r61051 = 1.0;
        double r61052 = r61050 + r61051;
        double r61053 = n;
        double r61054 = r61051 / r61053;
        double r61055 = pow(r61052, r61054);
        double r61056 = pow(r61050, r61054);
        double r61057 = r61055 - r61056;
        return r61057;
}

↓

double f(double x, double n) {
        double r61058 = n;
        double r61059 = -3406.4169782224208;
        bool r61060 = r61058 <= r61059;
        double r61061 = 1031392298459.7406;
        bool r61062 = r61058 <= r61061;
        double r61063 = !r61062;
        bool r61064 = r61060 || r61063;
        double r61065 = 1.0;
        double r61066 = x;
        double r61067 = r61065 / r61066;
        double r61068 = 1.0;
        double r61069 = r61068 / r61058;
        double r61070 = log(r61066);
        double r61071 = -r61070;
        double r61072 = 2.0;
        double r61073 = pow(r61058, r61072);
        double r61074 = r61071 / r61073;
        double r61075 = r61069 - r61074;
        double r61076 = r61067 * r61075;
        double r61077 = 0.5;
        double r61078 = pow(r61066, r61072);
        double r61079 = r61078 * r61058;
        double r61080 = r61077 / r61079;
        double r61081 = r61076 - r61080;
        double r61082 = r61066 + r61065;
        double r61083 = r61065 / r61058;
        double r61084 = r61083 / r61072;
        double r61085 = pow(r61082, r61084);
        double r61086 = pow(r61066, r61084);
        double r61087 = r61085 + r61086;
        double r61088 = r61085 - r61086;
        double r61089 = 3.0;
        double r61090 = pow(r61088, r61089);
        double r61091 = cbrt(r61090);
        double r61092 = pow(r61091, r61089);
        double r61093 = cbrt(r61092);
        double r61094 = pow(r61093, r61089);
        double r61095 = cbrt(r61094);
        double r61096 = r61087 * r61095;
        double r61097 = r61064 ? r61081 : r61096;
        return r61097;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -3406.4169782224208 or 1031392298459.7406 < n

   1. Initial program 44.9

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

   2. Taylor expanded around inf 33.1

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

   3. Simplified 32.5

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

   if -3406.4169782224208 < n < 1031392298459.7406

   1. Initial program 8.3

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

   3. Applied sqr-pow 8.3

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

   4. Applied sqr-pow 8.3

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

   5. Applied difference-of-squares 8.3

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

   7. Applied add-cbrt-cube 8.3

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

   8. Simplified 8.3

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

   10. Applied add-cbrt-cube 8.3

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

   11. Simplified 8.3

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

   13. Applied add-cbrt-cube 8.3

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

   14. Simplified 8.3

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3406.416978222420766542199999094009399414 \lor \neg \left(n \le 1031392298459.7406005859375\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))