2nthrt (problem 3.4.6)

Average Error: 33.0 → 23.9

Time: 24.6s

Precision: 64

• could not determine a ground truth for program body

  1. x = 10920358539895.18
  2. n = 9.278264258908537e-20

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.380908236352897 \cdot 10^{-15}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1245365987981562 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double n) {
        double r62986 = x;
        double r62987 = 1.0;
        double r62988 = r62986 + r62987;
        double r62989 = n;
        double r62990 = r62987 / r62989;
        double r62991 = pow(r62988, r62990);
        double r62992 = pow(r62986, r62990);
        double r62993 = r62991 - r62992;
        return r62993;
}

↓

double f(double x, double n) {
        double r62994 = 1.0;
        double r62995 = n;
        double r62996 = r62994 / r62995;
        double r62997 = -7.380908236352897e-15;
        bool r62998 = r62996 <= r62997;
        double r62999 = x;
        double r63000 = r62999 + r62994;
        double r63001 = pow(r63000, r62996);
        double r63002 = cbrt(r62994);
        double r63003 = r63002 * r63002;
        double r63004 = cbrt(r62995);
        double r63005 = r63004 * r63004;
        double r63006 = r63003 / r63005;
        double r63007 = pow(r62999, r63006);
        double r63008 = r63002 / r63004;
        double r63009 = pow(r63007, r63008);
        double r63010 = r63001 - r63009;
        double r63011 = 1.1245365987981562e-11;
        bool r63012 = r62996 <= r63011;
        double r63013 = r62996 / r62999;
        double r63014 = 0.5;
        double r63015 = r63014 / r62995;
        double r63016 = 2.0;
        double r63017 = pow(r62999, r63016);
        double r63018 = r63015 / r63017;
        double r63019 = log(r62999);
        double r63020 = r63019 * r62994;
        double r63021 = pow(r62995, r63016);
        double r63022 = r62999 * r63021;
        double r63023 = r63020 / r63022;
        double r63024 = r63018 - r63023;
        double r63025 = r63013 - r63024;
        double r63026 = cbrt(r63010);
        double r63027 = r63026 * r63026;
        double r63028 = r63027 * r63026;
        double r63029 = 3.0;
        double r63030 = pow(r63028, r63029);
        double r63031 = cbrt(r63030);
        double r63032 = r63012 ? r63025 : r63031;
        double r63033 = r62998 ? r63010 : r63032;
        return r63033;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -7.380908236352897e-15

   1. Initial program 3.3

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

   3. Applied add-cube-cbrt 3.4

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

   4. Applied add-cube-cbrt 3.4

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

   5. Applied times-frac 3.4

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

   6. Applied pow-unpow 3.4

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

   if -7.380908236352897e-15 < (/ 1.0 n) < 1.1245365987981562e-11

   1. Initial program 45.3

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

   2. Taylor expanded around inf 32.9

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

   3. Simplified 32.2

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

   if 1.1245365987981562e-11 < (/ 1.0 n)

   1. Initial program 7.6

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

   3. Applied add-cube-cbrt 7.7

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

   4. Applied add-cube-cbrt 7.7

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

   5. Applied times-frac 7.7

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

   6. Applied pow-unpow 7.6

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

   8. Applied add-cbrt-cube 7.7

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

   9. Simplified 7.7

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

   11. Applied add-cube-cbrt 7.7

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

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.380908236352897 \cdot 10^{-15}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 1.1245365987981562 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{n} \cdot \sqrt[3]{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{n}}\right)}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))