2nthrt (problem 3.4.6)

Average Error: 29.2 → 21.7

Time: 29.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.771691118882549e+275
  2. n = 2.9315320292795686e-164

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.001896405593671503979694148434020917193266:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.522757396775943809753988441961503053079 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double n) {
        double r61952 = x;
        double r61953 = 1.0;
        double r61954 = r61952 + r61953;
        double r61955 = n;
        double r61956 = r61953 / r61955;
        double r61957 = pow(r61954, r61956);
        double r61958 = pow(r61952, r61956);
        double r61959 = r61957 - r61958;
        return r61959;
}

↓

double f(double x, double n) {
        double r61960 = 1.0;
        double r61961 = n;
        double r61962 = r61960 / r61961;
        double r61963 = -0.001896405593671504;
        bool r61964 = r61962 <= r61963;
        double r61965 = x;
        double r61966 = r61965 + r61960;
        double r61967 = pow(r61966, r61962);
        double r61968 = pow(r61965, r61962);
        double r61969 = r61967 - r61968;
        double r61970 = cbrt(r61969);
        double r61971 = 6.0;
        double r61972 = pow(r61970, r61971);
        double r61973 = cbrt(r61972);
        double r61974 = r61973 * r61970;
        double r61975 = 3.0;
        double r61976 = pow(r61974, r61975);
        double r61977 = cbrt(r61976);
        double r61978 = pow(r61977, r61975);
        double r61979 = cbrt(r61978);
        double r61980 = 7.522757396775944e-09;
        bool r61981 = r61962 <= r61980;
        double r61982 = r61965 * r61961;
        double r61983 = r61960 / r61982;
        double r61984 = log(r61965);
        double r61985 = -r61984;
        double r61986 = 2.0;
        double r61987 = pow(r61961, r61986);
        double r61988 = r61965 * r61987;
        double r61989 = r61985 / r61988;
        double r61990 = 0.5;
        double r61991 = pow(r61965, r61986);
        double r61992 = r61991 * r61961;
        double r61993 = r61990 / r61992;
        double r61994 = fma(r61960, r61989, r61993);
        double r61995 = r61983 - r61994;
        double r61996 = sqrt(r61967);
        double r61997 = r61962 / r61986;
        double r61998 = pow(r61965, r61997);
        double r61999 = r61996 + r61998;
        double r62000 = r61996 - r61998;
        double r62001 = r61999 * r62000;
        double r62002 = cbrt(r62001);
        double r62003 = r61973 * r62002;
        double r62004 = pow(r62003, r61975);
        double r62005 = cbrt(r62004);
        double r62006 = pow(r62005, r61975);
        double r62007 = cbrt(r62006);
        double r62008 = r61981 ? r61995 : r62007;
        double r62009 = r61964 ? r61979 : r62008;
        return r62009;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.001896405593671504

   1. Initial program 0.2

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

   3. Applied add-cbrt-cube 0.2

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

   4. Simplified 0.2

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

   6. Applied add-cbrt-cube 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cube-cbrt 0.2

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

   10. Simplified 0.2

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\]

   if -0.001896405593671504 < (/ 1.0 n) < 7.522757396775944e-09

   1. Initial program 44.8

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

   2. Taylor expanded around inf 31.9

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

   3. Simplified 31.9

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

   if 7.522757396775944e-09 < (/ 1.0 n)

   1. Initial program 24.7

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

   3. Applied add-cbrt-cube 24.7

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

   4. Simplified 24.7

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

   6. Applied add-cbrt-cube 24.7

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

   7. Simplified 24.7

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

   9. Applied add-cube-cbrt 24.7

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

   10. Simplified 24.7

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\]
   11. Using strategy rm

   12. Applied sqr-pow 24.7

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)}^{3}}\right)}^{3}}\]

   13. Applied add-sqr-sqrt 24.7

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)}^{3}}\right)}^{3}}\]

   14. Applied difference-of-squares 24.7

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right)}^{3}}\right)}^{3}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 21.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.001896405593671503979694148434020917193266:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 7.522757396775943809753988441961503053079 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{6}} \cdot \sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))