2nthrt (problem 3.4.6)

Average Error: 29.2 → 19.1

Time: 30.7s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.9693088883133704e+180
  2. n = 4.0130259019332014e-262

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.267281974669497 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 7.385489355116676 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \left(-\frac{\frac{\log x}{x}}{n \cdot n}\right)\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r1492467 = x;
        double r1492468 = 1.0;
        double r1492469 = r1492467 + r1492468;
        double r1492470 = n;
        double r1492471 = r1492468 / r1492470;
        double r1492472 = pow(r1492469, r1492471);
        double r1492473 = pow(r1492467, r1492471);
        double r1492474 = r1492472 - r1492473;
        return r1492474;
}

↓

double f(double x, double n) {
        double r1492475 = 1.0;
        double r1492476 = n;
        double r1492477 = r1492475 / r1492476;
        double r1492478 = -6.267281974669497e-08;
        bool r1492479 = r1492477 <= r1492478;
        double r1492480 = x;
        double r1492481 = r1492480 + r1492475;
        double r1492482 = pow(r1492481, r1492477);
        double r1492483 = pow(r1492480, r1492477);
        double r1492484 = r1492482 - r1492483;
        double r1492485 = cbrt(r1492484);
        double r1492486 = sqrt(r1492482);
        double r1492487 = 2.0;
        double r1492488 = r1492477 / r1492487;
        double r1492489 = pow(r1492480, r1492488);
        double r1492490 = r1492486 - r1492489;
        double r1492491 = r1492489 + r1492486;
        double r1492492 = r1492490 * r1492491;
        double r1492493 = cbrt(r1492492);
        double r1492494 = cbrt(r1492480);
        double r1492495 = pow(r1492494, r1492477);
        double r1492496 = -r1492495;
        double r1492497 = r1492494 * r1492494;
        double r1492498 = pow(r1492497, r1492477);
        double r1492499 = r1492498 * r1492495;
        double r1492500 = fma(r1492496, r1492498, r1492499);
        double r1492501 = -r1492499;
        double r1492502 = fma(r1492475, r1492482, r1492501);
        double r1492503 = r1492500 + r1492502;
        double r1492504 = cbrt(r1492503);
        double r1492505 = r1492493 * r1492504;
        double r1492506 = r1492485 * r1492505;
        double r1492507 = 7.385489355116676e-07;
        bool r1492508 = r1492477 <= r1492507;
        double r1492509 = r1492480 * r1492476;
        double r1492510 = r1492475 / r1492509;
        double r1492511 = log(r1492480);
        double r1492512 = r1492511 / r1492480;
        double r1492513 = r1492476 * r1492476;
        double r1492514 = r1492512 / r1492513;
        double r1492515 = -r1492514;
        double r1492516 = r1492510 - r1492515;
        double r1492517 = 0.5;
        double r1492518 = r1492480 * r1492480;
        double r1492519 = r1492476 * r1492518;
        double r1492520 = r1492517 / r1492519;
        double r1492521 = r1492516 - r1492520;
        double r1492522 = log1p(r1492480);
        double r1492523 = r1492522 / r1492476;
        double r1492524 = exp(r1492523);
        double r1492525 = r1492524 - r1492483;
        double r1492526 = r1492508 ? r1492521 : r1492525;
        double r1492527 = r1492479 ? r1492506 : r1492526;
        return r1492527;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -6.267281974669497e-08

   1. Initial program 0.6

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

   3. Applied add-cube-cbrt 0.6

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

   5. Applied add-cube-cbrt 0.6

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

   6. Applied unpow-prod-down 0.6

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

   7. Applied *-un-lft-identity 0.6

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

   8. Applied prod-diff 0.6

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

   10. Applied sqr-pow 0.6

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

   11. Applied add-sqr-sqrt 0.6

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

   12. Applied difference-of-squares 0.6

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

   if -6.267281974669497e-08 < (/ 1 n) < 7.385489355116676e-07

   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 33.1

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

   3. Simplified 33.1

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

   if 7.385489355116676e-07 < (/ 1 n)

   1. Initial program 24.9

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

   3. Applied add-exp-log 24.9

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

   4. Applied pow-exp 24.9

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

   5. Simplified 0.9

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 19.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -6.267281974669497 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 7.385489355116676 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \left(-\frac{\frac{\log x}{x}}{n \cdot n}\right)\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))