2nthrt (problem 3.4.6)

Average Error: 29.6 → 23.1

Time: 33.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.4565393843068548e+200
  2. n = 1.837913291573934e-184

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 -4.895039802440850170454212278486686446129 \cdot 10^{-17}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.260859621565214258622089794815744532231 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{0.5}{n}}{x \cdot x}\right) + \frac{\log x \cdot 1}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r4670353 = x;
        double r4670354 = 1.0;
        double r4670355 = r4670353 + r4670354;
        double r4670356 = n;
        double r4670357 = r4670354 / r4670356;
        double r4670358 = pow(r4670355, r4670357);
        double r4670359 = pow(r4670353, r4670357);
        double r4670360 = r4670358 - r4670359;
        return r4670360;
}

↓

double f(double x, double n) {
        double r4670361 = 1.0;
        double r4670362 = n;
        double r4670363 = r4670361 / r4670362;
        double r4670364 = -4.89503980244085e-17;
        bool r4670365 = r4670363 <= r4670364;
        double r4670366 = x;
        double r4670367 = r4670366 + r4670361;
        double r4670368 = pow(r4670367, r4670363);
        double r4670369 = cbrt(r4670366);
        double r4670370 = r4670369 * r4670369;
        double r4670371 = pow(r4670370, r4670363);
        double r4670372 = cbrt(r4670369);
        double r4670373 = r4670372 * r4670372;
        double r4670374 = r4670372 * r4670373;
        double r4670375 = pow(r4670374, r4670363);
        double r4670376 = cbrt(r4670375);
        double r4670377 = r4670376 * r4670376;
        double r4670378 = r4670376 * r4670377;
        double r4670379 = r4670371 * r4670378;
        double r4670380 = r4670368 - r4670379;
        double r4670381 = cbrt(r4670380);
        double r4670382 = r4670381 * r4670381;
        double r4670383 = r4670381 * r4670382;
        double r4670384 = 1.2608596215652143e-44;
        bool r4670385 = r4670363 <= r4670384;
        double r4670386 = r4670366 * r4670362;
        double r4670387 = r4670361 / r4670386;
        double r4670388 = 0.5;
        double r4670389 = r4670388 / r4670362;
        double r4670390 = r4670366 * r4670366;
        double r4670391 = r4670389 / r4670390;
        double r4670392 = r4670387 - r4670391;
        double r4670393 = log(r4670366);
        double r4670394 = r4670393 * r4670361;
        double r4670395 = r4670362 * r4670362;
        double r4670396 = r4670395 * r4670366;
        double r4670397 = r4670394 / r4670396;
        double r4670398 = r4670392 + r4670397;
        double r4670399 = pow(r4670369, r4670363);
        double r4670400 = r4670371 * r4670399;
        double r4670401 = r4670368 - r4670400;
        double r4670402 = sqrt(r4670401);
        double r4670403 = r4670402 * r4670402;
        double r4670404 = r4670385 ? r4670398 : r4670403;
        double r4670405 = r4670365 ? r4670383 : r4670404;
        return r4670405;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -4.89503980244085e-17

   1. Initial program 2.2

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

   3. Applied add-cube-cbrt 2.2

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

   4. Applied unpow-prod-down 2.2

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

   6. Applied add-cube-cbrt 2.3

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

   8. Applied add-cube-cbrt 2.3

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

   10. Applied add-cube-cbrt 2.3

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

   if -4.89503980244085e-17 < (/ 1.0 n) < 1.2608596215652143e-44

   1. Initial program 44.5

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

   2. Taylor expanded around inf 32.2

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

   3. Simplified 32.2

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

   if 1.2608596215652143e-44 < (/ 1.0 n)

   1. Initial program 30.5

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

   3. Applied add-cube-cbrt 30.5

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

   4. Applied unpow-prod-down 30.6

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

   6. Applied add-sqr-sqrt 30.6

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

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4.895039802440850170454212278486686446129 \cdot 10^{-17}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)} \cdot \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)}}\right)\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.260859621565214258622089794815744532231 \cdot 10^{-44}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{\frac{0.5}{n}}{x \cdot x}\right) + \frac{\log x \cdot 1}{\left(n \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))