2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.9

Time: 28.7s

Precision: 64

• could not determine a ground truth for program body

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.4291090181189753 \cdot 10^{-7}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.97263161662073162 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{2}}\right)}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r75471 = x;
        double r75472 = 1.0;
        double r75473 = r75471 + r75472;
        double r75474 = n;
        double r75475 = r75472 / r75474;
        double r75476 = pow(r75473, r75475);
        double r75477 = pow(r75471, r75475);
        double r75478 = r75476 - r75477;
        return r75478;
}

↓

double f(double x, double n) {
        double r75479 = 1.0;
        double r75480 = n;
        double r75481 = r75479 / r75480;
        double r75482 = -1.4291090181189753e-07;
        bool r75483 = r75481 <= r75482;
        double r75484 = x;
        double r75485 = r75484 + r75479;
        double r75486 = 2.0;
        double r75487 = r75481 / r75486;
        double r75488 = pow(r75485, r75487);
        double r75489 = pow(r75484, r75487);
        double r75490 = log1p(r75489);
        double r75491 = expm1(r75490);
        double r75492 = r75488 + r75491;
        double r75493 = cbrt(r75484);
        double r75494 = r75493 * r75493;
        double r75495 = pow(r75494, r75487);
        double r75496 = pow(r75493, r75487);
        double r75497 = r75495 * r75496;
        double r75498 = r75488 - r75497;
        double r75499 = r75492 * r75498;
        double r75500 = 4.972631616620732e-22;
        bool r75501 = r75481 <= r75500;
        double r75502 = r75484 * r75480;
        double r75503 = r75479 / r75502;
        double r75504 = log(r75484);
        double r75505 = -r75504;
        double r75506 = pow(r75480, r75486);
        double r75507 = r75484 * r75506;
        double r75508 = r75505 / r75507;
        double r75509 = 0.5;
        double r75510 = pow(r75484, r75486);
        double r75511 = r75510 * r75480;
        double r75512 = r75509 / r75511;
        double r75513 = fma(r75479, r75508, r75512);
        double r75514 = r75503 - r75513;
        double r75515 = 1.0;
        double r75516 = cbrt(r75480);
        double r75517 = r75516 * r75516;
        double r75518 = r75515 / r75517;
        double r75519 = sqrt(r75486);
        double r75520 = r75518 / r75519;
        double r75521 = pow(r75484, r75520);
        double r75522 = r75479 / r75516;
        double r75523 = r75522 / r75519;
        double r75524 = pow(r75521, r75523);
        double r75525 = r75488 - r75524;
        double r75526 = r75492 * r75525;
        double r75527 = r75501 ? r75514 : r75526;
        double r75528 = r75483 ? r75499 : r75527;
        return r75528;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.8

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

   3. Applied sqr-pow 0.8

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

   4. Applied sqr-pow 0.8

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

   5. Applied difference-of-squares 0.8

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

   7. Applied expm1-log1p-u 0.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 0.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]

   10. Applied unpow-prod-down 0.8

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \color{blue}{{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}}\right)\]

   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. Taylor expanded around inf 32.9

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

   3. Simplified 32.9

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

   if 4.972631616620732e-22 < (/ 1.0 n)

   1. Initial program 26.8

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

   3. Applied sqr-pow 26.9

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

   4. Applied sqr-pow 26.8

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

   5. Applied difference-of-squares 26.8

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

   7. Applied expm1-log1p-u 26.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 26.8

      \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}\right)}\right)\]

   10. Applied add-cube-cbrt 26.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{\color{blue}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}}{\sqrt{2} \cdot \sqrt{2}}\right)}\right)\]

   11. Applied *-un-lft-identity 26.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}}}{\sqrt{2} \cdot \sqrt{2}}\right)}\right)\]

   12. Applied times-frac 26.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\color{blue}{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}} \cdot \frac{1}{\sqrt[3]{n}}}}{\sqrt{2} \cdot \sqrt{2}}\right)}\right)\]

   13. Applied times-frac 26.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\color{blue}{\left(\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{2}} \cdot \frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{2}}\right)}}\right)\]

   14. Applied pow-unpow 26.9

       \[\leadsto \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \color{blue}{{\left({x}^{\left(\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{2}}\right)}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 22.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.4291090181189753 \cdot 10^{-7}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 4.97263161662073162 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(1, \frac{-\log x}{x \cdot {n}^{2}}, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \mathsf{expm1}\left(\mathsf{log1p}\left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left({x}^{\left(\frac{\frac{1}{\sqrt[3]{n} \cdot \sqrt[3]{n}}}{\sqrt{2}}\right)}\right)}^{\left(\frac{\frac{1}{\sqrt[3]{n}}}{\sqrt{2}}\right)}\right)\\ \end{array}\]

Reproduce

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