2nthrt (problem 3.4.6)

Average Error: 29.5 → 23.0

Time: 34.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.2545226427089226e+284
  2. n = 1.8034259511601163e-130

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.05925712577091127308825946329307043924928:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.255271015860772358417468981097305856376 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\\ \end{array}\]

C

double f(double x, double n) {
        double r67918 = x;
        double r67919 = 1.0;
        double r67920 = r67918 + r67919;
        double r67921 = n;
        double r67922 = r67919 / r67921;
        double r67923 = pow(r67920, r67922);
        double r67924 = pow(r67918, r67922);
        double r67925 = r67923 - r67924;
        return r67925;
}

↓

double f(double x, double n) {
        double r67926 = 1.0;
        double r67927 = n;
        double r67928 = r67926 / r67927;
        double r67929 = -0.05925712577091127;
        bool r67930 = r67928 <= r67929;
        double r67931 = x;
        double r67932 = r67931 + r67926;
        double r67933 = pow(r67932, r67928);
        double r67934 = sqrt(r67933);
        double r67935 = 2.0;
        double r67936 = r67928 / r67935;
        double r67937 = pow(r67931, r67936);
        double r67938 = r67934 + r67937;
        double r67939 = r67934 - r67937;
        double r67940 = r67938 * r67939;
        double r67941 = cbrt(r67940);
        double r67942 = pow(r67931, r67928);
        double r67943 = r67933 - r67942;
        double r67944 = cbrt(r67943);
        double r67945 = r67941 * r67944;
        double r67946 = pow(r67932, r67936);
        double r67947 = sqrt(r67942);
        double r67948 = r67946 + r67947;
        double r67949 = r67946 - r67947;
        double r67950 = r67948 * r67949;
        double r67951 = cbrt(r67950);
        double r67952 = r67945 * r67951;
        double r67953 = 2.2552710158607724e-27;
        bool r67954 = r67928 <= r67953;
        double r67955 = r67926 / r67931;
        double r67956 = 1.0;
        double r67957 = r67956 / r67927;
        double r67958 = log(r67931);
        double r67959 = -r67958;
        double r67960 = pow(r67927, r67935);
        double r67961 = r67959 / r67960;
        double r67962 = r67957 - r67961;
        double r67963 = r67955 * r67962;
        double r67964 = 0.5;
        double r67965 = pow(r67931, r67935);
        double r67966 = r67965 * r67927;
        double r67967 = r67964 / r67966;
        double r67968 = r67963 - r67967;
        double r67969 = cbrt(r67939);
        double r67970 = r67969 * r67969;
        double r67971 = r67970 * r67969;
        double r67972 = r67938 * r67971;
        double r67973 = cbrt(r67972);
        double r67974 = r67973 * r67944;
        double r67975 = r67944 * r67944;
        double r67976 = r67975 * r67944;
        double r67977 = cbrt(r67976);
        double r67978 = r67974 * r67977;
        double r67979 = r67954 ? r67968 : r67978;
        double r67980 = r67930 ? r67952 : r67979;
        return r67980;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.05925712577091127

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   5. Applied sqr-pow 0.1

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

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied difference-of-squares 0.1

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

   9. Applied add-sqr-sqrt 0.1

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

   10. Applied sqr-pow 0.1

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

   11. Applied difference-of-squares 0.1

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

   if -0.05925712577091127 < (/ 1.0 n) < 2.2552710158607724e-27

   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 33.9

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

   3. Simplified 33.3

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

   if 2.2552710158607724e-27 < (/ 1.0 n)

   1. Initial program 28.1

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

   3. Applied add-cube-cbrt 28.2

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

   5. Applied sqr-pow 28.1

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

   6. Applied add-sqr-sqrt 28.1

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

   7. Applied difference-of-squares 28.1

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

   9. Applied add-cube-cbrt 28.1

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

   11. Applied add-cube-cbrt 28.1

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

4. Final simplification 23.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.05925712577091127308825946329307043924928:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.255271015860772358417468981097305856376 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))