2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.3

Time: 16.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 4.773292077298795e+235
  2. n = 2.4612524637435313e-181

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - 1 \cdot \frac{-\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

C

double f(double x, double n) {
        double r67618 = x;
        double r67619 = 1.0;
        double r67620 = r67618 + r67619;
        double r67621 = n;
        double r67622 = r67619 / r67621;
        double r67623 = pow(r67620, r67622);
        double r67624 = pow(r67618, r67622);
        double r67625 = r67623 - r67624;
        return r67625;
}

↓

double f(double x, double n) {
        double r67626 = 1.0;
        double r67627 = n;
        double r67628 = r67626 / r67627;
        double r67629 = -2.429908229671821e-07;
        bool r67630 = r67628 <= r67629;
        double r67631 = x;
        double r67632 = r67631 + r67626;
        double r67633 = pow(r67632, r67628);
        double r67634 = pow(r67631, r67628);
        double r67635 = r67633 - r67634;
        double r67636 = 3.0;
        double r67637 = pow(r67635, r67636);
        double r67638 = cbrt(r67637);
        double r67639 = pow(r67638, r67636);
        double r67640 = cbrt(r67639);
        double r67641 = pow(r67640, r67636);
        double r67642 = cbrt(r67641);
        double r67643 = pow(r67642, r67636);
        double r67644 = cbrt(r67643);
        double r67645 = 1.4807923677373035e-15;
        bool r67646 = r67628 <= r67645;
        double r67647 = r67631 * r67627;
        double r67648 = r67626 / r67647;
        double r67649 = log(r67631);
        double r67650 = -r67649;
        double r67651 = 2.0;
        double r67652 = pow(r67627, r67651);
        double r67653 = r67631 * r67652;
        double r67654 = r67650 / r67653;
        double r67655 = r67626 * r67654;
        double r67656 = r67648 - r67655;
        double r67657 = 0.5;
        double r67658 = pow(r67631, r67651);
        double r67659 = r67658 * r67627;
        double r67660 = r67657 / r67659;
        double r67661 = r67656 - r67660;
        double r67662 = sqrt(r67635);
        double r67663 = r67662 * r67662;
        double r67664 = pow(r67663, r67636);
        double r67665 = cbrt(r67664);
        double r67666 = pow(r67665, r67636);
        double r67667 = cbrt(r67666);
        double r67668 = r67646 ? r67661 : r67667;
        double r67669 = r67630 ? r67644 : r67668;
        return r67669;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -2.429908229671821e-07

   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-cbrt-cube 0.6

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

   4. Simplified 0.6

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

   6. Applied add-cbrt-cube 0.6

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

   7. Simplified 0.6

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

   9. Applied add-cbrt-cube 0.6

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

   10. Simplified 0.6

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

   12. Applied add-cbrt-cube 0.6

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

   13. Simplified 0.6

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

   if -2.429908229671821e-07 < (/ 1.0 n) < 1.4807923677373035e-15

   1. Initial program 44.9

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

   3. Applied add-cbrt-cube 44.9

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

   4. Simplified 44.9

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

   5. Taylor expanded around inf 32.2

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

   6. Simplified 32.2

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

   if 1.4807923677373035e-15 < (/ 1.0 n)

   1. Initial program 27.2

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

   3. Applied add-cbrt-cube 27.2

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

   4. Simplified 27.2

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

   6. Applied add-cbrt-cube 27.2

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

   7. Simplified 27.2

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

   9. Applied add-sqr-sqrt 27.2

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

4. Final simplification 22.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - 1 \cdot \frac{-\log x}{x \cdot {n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}^{3}}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))