2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.8

Time: 17.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.60915386116327e+88
  2. n = 1.8004234423016248e-156

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4044640181538.08936:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\mathsf{fma}\left({\left(\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, {\left(\frac{1}{x}\right)}^{\frac{-1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.46973031667613272 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \frac{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x + 1}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r112398 = x;
        double r112399 = 1.0;
        double r112400 = r112398 + r112399;
        double r112401 = n;
        double r112402 = r112399 / r112401;
        double r112403 = pow(r112400, r112402);
        double r112404 = pow(r112398, r112402);
        double r112405 = r112403 - r112404;
        return r112405;
}

↓

double f(double x, double n) {
        double r112406 = 1.0;
        double r112407 = n;
        double r112408 = r112406 / r112407;
        double r112409 = -4044640181538.0894;
        bool r112410 = r112408 <= r112409;
        double r112411 = x;
        double r112412 = r112411 + r112406;
        double r112413 = sqrt(r112412);
        double r112414 = pow(r112413, r112408);
        double r112415 = 2.0;
        double r112416 = r112408 / r112415;
        double r112417 = pow(r112411, r112416);
        double r112418 = r112414 + r112417;
        double r112419 = 1.0;
        double r112420 = pow(r112411, r112415);
        double r112421 = r112419 / r112420;
        double r112422 = 0.3333333333333333;
        double r112423 = pow(r112421, r112422);
        double r112424 = 0.3333333333333333;
        double r112425 = r112419 / r112411;
        double r112426 = -0.3333333333333333;
        double r112427 = pow(r112425, r112426);
        double r112428 = 0.1111111111111111;
        double r112429 = 5.0;
        double r112430 = pow(r112411, r112429);
        double r112431 = r112419 / r112430;
        double r112432 = pow(r112431, r112422);
        double r112433 = r112428 * r112432;
        double r112434 = r112427 - r112433;
        double r112435 = fma(r112423, r112424, r112434);
        double r112436 = pow(r112435, r112408);
        double r112437 = cbrt(r112413);
        double r112438 = pow(r112437, r112408);
        double r112439 = sqrt(r112411);
        double r112440 = pow(r112439, r112416);
        double r112441 = r112440 * r112440;
        double r112442 = -r112441;
        double r112443 = fma(r112436, r112438, r112442);
        double r112444 = -r112440;
        double r112445 = r112415 * r112416;
        double r112446 = pow(r112439, r112445);
        double r112447 = fma(r112440, r112444, r112446);
        double r112448 = r112443 + r112447;
        double r112449 = r112418 * r112448;
        double r112450 = 1.4697303166761327e-07;
        bool r112451 = r112408 <= r112450;
        double r112452 = r112411 * r112407;
        double r112453 = r112419 / r112452;
        double r112454 = 0.5;
        double r112455 = r112420 * r112407;
        double r112456 = r112419 / r112455;
        double r112457 = log(r112425);
        double r112458 = pow(r112407, r112415);
        double r112459 = r112411 * r112458;
        double r112460 = r112457 / r112459;
        double r112461 = r112406 * r112460;
        double r112462 = fma(r112454, r112456, r112461);
        double r112463 = -r112462;
        double r112464 = fma(r112406, r112453, r112463);
        double r112465 = 3.0;
        double r112466 = pow(r112414, r112465);
        double r112467 = pow(r112417, r112465);
        double r112468 = r112466 - r112467;
        double r112469 = r112415 * r112408;
        double r112470 = pow(r112413, r112469);
        double r112471 = fma(r112417, r112418, r112470);
        double r112472 = r112468 / r112471;
        double r112473 = r112418 * r112472;
        double r112474 = r112451 ? r112464 : r112473;
        double r112475 = r112410 ? r112449 : r112474;
        return r112475;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -4044640181538.0894

   1. Initial program 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

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

   4. Applied add-sqr-sqrt 0

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

   5. Applied unpow-prod-down 0

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

   6. Applied difference-of-squares 0

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

   8. Applied add-sqr-sqrt 0

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

   9. Applied unpow-prod-down 0

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

   10. Applied add-cube-cbrt 0

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

   11. Applied unpow-prod-down 0

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

   12. Applied prod-diff 0

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

   13. Simplified 0

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

   14. Taylor expanded around inf 0

       \[\leadsto \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\mathsf{fma}\left({\color{blue}{\left(\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)\]

   15. Simplified 0

       \[\leadsto \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\mathsf{fma}\left({\color{blue}{\left(\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, {\left(\frac{1}{x}\right)}^{\frac{-1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\right)}}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)\]

   if -4044640181538.0894 < (/ 1.0 n) < 1.4697303166761327e-07

   1. Initial program 44.4

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

   2. Taylor expanded around inf 33.2

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

   3. Simplified 33.2

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

   if 1.4697303166761327e-07 < (/ 1.0 n)

   1. Initial program 6.5

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

   3. Applied sqr-pow 6.6

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

   4. Applied add-sqr-sqrt 6.6

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

   5. Applied unpow-prod-down 6.5

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

   6. Applied difference-of-squares 6.5

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

   8. Applied flip3-- 6.5

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

   9. Simplified 6.5

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

4. Final simplification 24.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -4044640181538.08936:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\mathsf{fma}\left({\left(\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, {\left(\frac{1}{x}\right)}^{\frac{-1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{\sqrt{x + 1}}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{\frac{1}{n}}{2}\right)}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.46973031667613272 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{x \cdot n}, -\mathsf{fma}\left(0.5, \frac{1}{{x}^{2} \cdot n}, 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \frac{{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}^{3}}{\mathsf{fma}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}, {\left(\sqrt{x + 1}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))