2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.7

Time: 28.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.6197547301821798e+147
  2. n = 4.2985835605462347e-277

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.4376991383250568024010362933040596544743:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.823892541888928748751268970584802409735 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt[3]{\sqrt{2}}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r81977 = x;
        double r81978 = 1.0;
        double r81979 = r81977 + r81978;
        double r81980 = n;
        double r81981 = r81978 / r81980;
        double r81982 = pow(r81979, r81981);
        double r81983 = pow(r81977, r81981);
        double r81984 = r81982 - r81983;
        return r81984;
}

↓

double f(double x, double n) {
        double r81985 = 1.0;
        double r81986 = n;
        double r81987 = r81985 / r81986;
        double r81988 = -0.4376991383250568;
        bool r81989 = r81987 <= r81988;
        double r81990 = x;
        double r81991 = r81990 + r81985;
        double r81992 = 2.0;
        double r81993 = r81987 / r81992;
        double r81994 = pow(r81991, r81993);
        double r81995 = pow(r81990, r81993);
        double r81996 = r81994 + r81995;
        double r81997 = cbrt(r81987);
        double r81998 = r81997 * r81997;
        double r81999 = sqrt(r81992);
        double r82000 = r81998 / r81999;
        double r82001 = pow(r81990, r82000);
        double r82002 = r81997 / r81999;
        double r82003 = pow(r82001, r82002);
        double r82004 = r81994 - r82003;
        double r82005 = r81996 * r82004;
        double r82006 = 1.8238925418889287e-19;
        bool r82007 = r81987 <= r82006;
        double r82008 = r81985 / r81990;
        double r82009 = 1.0;
        double r82010 = r82009 / r81986;
        double r82011 = log(r81990);
        double r82012 = -r82011;
        double r82013 = pow(r81986, r81992);
        double r82014 = r82012 / r82013;
        double r82015 = r82010 - r82014;
        double r82016 = r82008 * r82015;
        double r82017 = 0.5;
        double r82018 = pow(r81990, r81992);
        double r82019 = r82018 * r81986;
        double r82020 = r82017 / r82019;
        double r82021 = r82016 - r82020;
        double r82022 = cbrt(r81999);
        double r82023 = r82022 * r82022;
        double r82024 = r81997 / r82023;
        double r82025 = pow(r81990, r82024);
        double r82026 = r81997 / r82022;
        double r82027 = pow(r82025, r82026);
        double r82028 = pow(r82027, r82002);
        double r82029 = r81994 - r82028;
        double r82030 = r81996 * r82029;
        double r82031 = r82007 ? r82021 : r82030;
        double r82032 = r81989 ? r82005 : r82031;
        return r82032;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.4376991383250568

   1. Initial program 0.0

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

   3. Applied sqr-pow 0.0

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

   4. Applied sqr-pow 0.0

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

   5. Applied difference-of-squares 0.0

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

   7. Applied add-sqr-sqrt 0.0

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

   8. Applied add-cube-cbrt 0.0

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

   9. Applied times-frac 0.0

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

   10. Applied pow-unpow 0.0

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

   if -0.4376991383250568 < (/ 1.0 n) < 1.8238925418889287e-19

   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 33.5

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

   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 1.8238925418889287e-19 < (/ 1.0 n)

   1. Initial program 28.3

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

   3. Applied sqr-pow 28.5

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

   4. Applied sqr-pow 28.4

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

   5. Applied difference-of-squares 28.4

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

   7. Applied add-sqr-sqrt 28.4

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

   8. Applied add-cube-cbrt 28.4

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

   9. Applied times-frac 28.4

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

   10. Applied pow-unpow 28.4

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

   12. Applied add-cube-cbrt 28.4

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

   13. Applied times-frac 28.4

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

   14. Applied pow-unpow 28.6

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

4. Final simplification 22.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.4376991383250568024010362933040596544743:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.823892541888928748751268970584802409735 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({\left({x}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt[3]{\sqrt{2}}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{\sqrt{2}}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))