2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.4

Time: 30.0s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.41496557713783e+117
  2. n = 1.3363806826620586e-187

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -380251775535541.9375 \lor \neg \left(n \le 0.9298484945110256916223079315386712551117\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\sqrt[3]{-\log x} \cdot \sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{4}} \cdot \frac{\sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r56713 = x;
        double r56714 = 1.0;
        double r56715 = r56713 + r56714;
        double r56716 = n;
        double r56717 = r56714 / r56716;
        double r56718 = pow(r56715, r56717);
        double r56719 = pow(r56713, r56717);
        double r56720 = r56718 - r56719;
        return r56720;
}

↓

double f(double x, double n) {
        double r56721 = n;
        double r56722 = -380251775535541.94;
        bool r56723 = r56721 <= r56722;
        double r56724 = 0.9298484945110257;
        bool r56725 = r56721 <= r56724;
        double r56726 = !r56725;
        bool r56727 = r56723 || r56726;
        double r56728 = 1.0;
        double r56729 = x;
        double r56730 = r56728 / r56729;
        double r56731 = 1.0;
        double r56732 = r56731 / r56721;
        double r56733 = log(r56729);
        double r56734 = -r56733;
        double r56735 = cbrt(r56734);
        double r56736 = r56735 * r56735;
        double r56737 = cbrt(r56721);
        double r56738 = 4.0;
        double r56739 = pow(r56737, r56738);
        double r56740 = r56736 / r56739;
        double r56741 = 2.0;
        double r56742 = pow(r56737, r56741);
        double r56743 = r56735 / r56742;
        double r56744 = r56740 * r56743;
        double r56745 = r56732 - r56744;
        double r56746 = r56730 * r56745;
        double r56747 = 0.5;
        double r56748 = pow(r56729, r56741);
        double r56749 = r56748 * r56721;
        double r56750 = r56747 / r56749;
        double r56751 = r56746 - r56750;
        double r56752 = r56729 + r56728;
        double r56753 = r56728 / r56721;
        double r56754 = pow(r56752, r56753);
        double r56755 = cbrt(r56729);
        double r56756 = r56755 * r56755;
        double r56757 = cbrt(r56756);
        double r56758 = cbrt(r56755);
        double r56759 = r56755 * r56758;
        double r56760 = r56757 * r56759;
        double r56761 = pow(r56760, r56753);
        double r56762 = pow(r56755, r56753);
        double r56763 = r56761 * r56762;
        double r56764 = r56754 - r56763;
        double r56765 = r56727 ? r56751 : r56764;
        return r56765;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if n < -380251775535541.94 or 0.9298484945110257 < n

   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

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

   3. Simplified 32.5

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

   5. Applied add-cube-cbrt 32.5

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

   6. Applied unpow-prod-down 32.5

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

   7. Applied add-cube-cbrt 32.5

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

   8. Applied times-frac 32.5

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

   9. Simplified 32.5

      \[\leadsto \frac{1}{x} \cdot \left(\frac{1}{n} - \color{blue}{\frac{\sqrt[3]{-\log x} \cdot \sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{4}}} \cdot \frac{\sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\]

   if -380251775535541.94 < n < 0.9298484945110257

   1. Initial program 9.1

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

   3. Applied add-cube-cbrt 9.1

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

   4. Applied unpow-prod-down 9.1

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied cbrt-prod 9.1

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

   8. Applied associate-*l* 9.1

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

   9. Simplified 9.1

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

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -380251775535541.9375 \lor \neg \left(n \le 0.9298484945110256916223079315386712551117\right):\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\sqrt[3]{-\log x} \cdot \sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{4}} \cdot \frac{\sqrt[3]{-\log x}}{{\left(\sqrt[3]{n}\right)}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))