2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.9

Time: 29.6s

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.982081272343965e+98
  2. n = 5.07224517664338e-55

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.4291090181189753 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 4.97263161662073162 \cdot 10^{-22}\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r68485 = x;
        double r68486 = 1.0;
        double r68487 = r68485 + r68486;
        double r68488 = n;
        double r68489 = r68486 / r68488;
        double r68490 = pow(r68487, r68489);
        double r68491 = pow(r68485, r68489);
        double r68492 = r68490 - r68491;
        return r68492;
}

↓

double f(double x, double n) {
        double r68493 = 1.0;
        double r68494 = n;
        double r68495 = r68493 / r68494;
        double r68496 = -1.4291090181189753e-07;
        bool r68497 = r68495 <= r68496;
        double r68498 = 4.972631616620732e-22;
        bool r68499 = r68495 <= r68498;
        double r68500 = !r68499;
        bool r68501 = r68497 || r68500;
        double r68502 = x;
        double r68503 = r68502 + r68493;
        double r68504 = 2.0;
        double r68505 = r68495 / r68504;
        double r68506 = pow(r68503, r68505);
        double r68507 = pow(r68502, r68505);
        double r68508 = r68506 + r68507;
        double r68509 = cbrt(r68495);
        double r68510 = r68509 * r68509;
        double r68511 = pow(r68502, r68510);
        double r68512 = r68509 / r68504;
        double r68513 = pow(r68511, r68512);
        double r68514 = r68506 - r68513;
        double r68515 = r68508 * r68514;
        double r68516 = 0.5;
        double r68517 = r68502 * r68494;
        double r68518 = r68516 / r68517;
        double r68519 = 0.25;
        double r68520 = 1.0;
        double r68521 = pow(r68502, r68504);
        double r68522 = r68521 * r68494;
        double r68523 = r68520 / r68522;
        double r68524 = log(r68502);
        double r68525 = -r68524;
        double r68526 = pow(r68494, r68504);
        double r68527 = r68502 * r68526;
        double r68528 = r68525 / r68527;
        double r68529 = r68523 + r68528;
        double r68530 = r68519 * r68529;
        double r68531 = r68518 - r68530;
        double r68532 = r68508 * r68531;
        double r68533 = r68501 ? r68515 : r68532;
        return r68533;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 n) < -1.4291090181189753e-07 or 4.972631616620732e-22 < (/ 1.0 n)

   1. Initial program 10.0

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

   3. Applied sqr-pow 10.1

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

   4. Applied sqr-pow 10.0

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

   5. Applied difference-of-squares 10.0

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

   7. Applied *-un-lft-identity 10.0

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

   8. Applied add-cube-cbrt 10.1

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

   9. Applied times-frac 10.1

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

   10. Applied pow-unpow 10.1

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

   11. Simplified 10.1

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

   if -1.4291090181189753e-07 < (/ 1.0 n) < 4.972631616620732e-22

   1. Initial program 44.7

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

   3. Applied sqr-pow 44.7

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

   4. Applied sqr-pow 44.7

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

   5. Applied difference-of-squares 44.7

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

   7. Applied *-un-lft-identity 44.7

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

   8. Applied add-cube-cbrt 44.7

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

   9. Applied times-frac 44.7

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

   10. Applied pow-unpow 44.7

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

   11. Simplified 44.7

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

   12. Taylor expanded around inf 32.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

   13. Simplified 32.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 22.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.4291090181189753 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 4.97263161662073162 \cdot 10^{-22}\right):\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\sqrt[3]{\frac{1}{n}} \cdot \sqrt[3]{\frac{1}{n}}\right)}\right)}^{\left(\frac{\sqrt[3]{\frac{1}{n}}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{-\log x}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))