2nthrt (problem 3.4.6)

Average Error: 33.2 → 23.6

Time: 16.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.531334608873196e+226
  2. n = 1.3050215705613157e-129

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.4934594501765746 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.780710866682253 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{{n}^{2}} \cdot \left(\frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x}\right) + \frac{1}{n} \cdot \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r80448 = x;
        double r80449 = 1.0;
        double r80450 = r80448 + r80449;
        double r80451 = n;
        double r80452 = r80449 / r80451;
        double r80453 = pow(r80450, r80452);
        double r80454 = pow(r80448, r80452);
        double r80455 = r80453 - r80454;
        return r80455;
}

↓

double f(double x, double n) {
        double r80456 = 1.0;
        double r80457 = n;
        double r80458 = r80456 / r80457;
        double r80459 = -2.4934594501765746e-07;
        bool r80460 = r80458 <= r80459;
        double r80461 = x;
        double r80462 = r80461 + r80456;
        double r80463 = pow(r80462, r80458);
        double r80464 = sqrt(r80461);
        double r80465 = 2.0;
        double r80466 = r80465 * r80458;
        double r80467 = pow(r80464, r80466);
        double r80468 = r80463 - r80467;
        double r80469 = exp(r80468);
        double r80470 = log(r80469);
        double r80471 = pow(r80464, r80458);
        double r80472 = sqrt(r80464);
        double r80473 = pow(r80472, r80458);
        double r80474 = r80473 * r80473;
        double r80475 = -r80474;
        double r80476 = fma(r80471, r80475, r80467);
        double r80477 = r80470 + r80476;
        double r80478 = 1.780710866682253e-05;
        bool r80479 = r80458 <= r80478;
        double r80480 = pow(r80457, r80465);
        double r80481 = r80456 / r80480;
        double r80482 = 1.0;
        double r80483 = r80482 / r80461;
        double r80484 = -0.3333333333333333;
        double r80485 = pow(r80483, r80484);
        double r80486 = log(r80485);
        double r80487 = r80486 / r80461;
        double r80488 = -0.6666666666666666;
        double r80489 = pow(r80483, r80488);
        double r80490 = log(r80489);
        double r80491 = r80490 / r80461;
        double r80492 = r80487 + r80491;
        double r80493 = r80481 * r80492;
        double r80494 = r80482 / r80457;
        double r80495 = r80456 / r80461;
        double r80496 = 0.5;
        double r80497 = pow(r80461, r80465);
        double r80498 = r80496 / r80497;
        double r80499 = r80495 - r80498;
        double r80500 = r80494 * r80499;
        double r80501 = r80493 + r80500;
        double r80502 = cbrt(r80461);
        double r80503 = r80502 * r80502;
        double r80504 = pow(r80503, r80458);
        double r80505 = pow(r80502, r80458);
        double r80506 = -r80505;
        double r80507 = r80506 + r80505;
        double r80508 = r80504 * r80507;
        double r80509 = r80501 + r80508;
        double r80510 = 3.0;
        double r80511 = pow(r80463, r80510);
        double r80512 = pow(r80467, r80510);
        double r80513 = r80511 - r80512;
        double r80514 = r80463 + r80467;
        double r80515 = pow(r80462, r80466);
        double r80516 = fma(r80467, r80514, r80515);
        double r80517 = r80513 / r80516;
        double r80518 = -r80471;
        double r80519 = fma(r80471, r80518, r80467);
        double r80520 = r80517 + r80519;
        double r80521 = r80479 ? r80509 : r80520;
        double r80522 = r80460 ? r80477 : r80521;
        return r80522;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -2.4934594501765746e-07

   1. Initial program 1.6

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

   3. Applied add-sqr-sqrt 1.6

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

   4. Applied unpow-prod-down 1.7

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

   5. Applied *-un-lft-identity 1.7

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

   6. Applied unpow-prod-down 1.7

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

   7. Applied prod-diff 1.7

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

   8. Simplified 1.7

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

   9. Simplified 1.7

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

   11. Applied add-sqr-sqrt 1.7

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

   12. Applied sqrt-prod 1.7

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

   13. Applied unpow-prod-down 1.7

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

   15. Applied add-log-exp 1.9

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

   16. Applied add-log-exp 1.9

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

   17. Applied diff-log 1.9

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

   18. Simplified 1.9

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

   if -2.4934594501765746e-07 < (/ 1.0 n) < 1.780710866682253e-05

   1. Initial program 44.8

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

   3. Applied add-cube-cbrt 44.8

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

   4. Applied unpow-prod-down 44.8

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

   5. Applied add-cube-cbrt 44.8

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

   6. Applied unpow-prod-down 44.8

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

   7. Applied prod-diff 44.8

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

   8. Simplified 44.8

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

   9. Taylor expanded around inf 32.0

      \[\leadsto \color{blue}{\left(\left(1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(1 \cdot \frac{1}{x \cdot n} + 1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   10. Simplified 31.4

       \[\leadsto \color{blue}{\left(\frac{1}{{n}^{2}} \cdot \left(\frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x}\right) + \frac{1}{n} \cdot \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   if 1.780710866682253e-05 < (/ 1.0 n)

   1. Initial program 5.1

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

   3. Applied add-sqr-sqrt 5.1

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

   4. Applied unpow-prod-down 5.1

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

   5. Applied *-un-lft-identity 5.1

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

   6. Applied unpow-prod-down 5.1

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

   7. Applied prod-diff 5.2

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

   8. Simplified 5.2

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

   9. Simplified 5.2

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

   11. Applied flip3-- 5.2

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

   12. Simplified 5.2

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

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.4934594501765746 \cdot 10^{-7}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}}\right) + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt{\sqrt{x}}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.780710866682253 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{{n}^{2}} \cdot \left(\frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x} + \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x}\right) + \frac{1}{n} \cdot \left(\frac{1}{x} - \frac{0.5}{{x}^{2}}\right)\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)}\right)}^{3} - {\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)}^{3}}{\mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}, {\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)} + \mathsf{fma}\left({\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt{x}\right)}^{\left(2 \cdot \frac{1}{n}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))