2nthrt (problem 3.4.6)

Average Error: 29.5 → 19.2

Time: 32.4s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.250954683344369e+182
  2. n = 2.0610646524110457e-176

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1982989920422248.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;n \le -4.76046101311837 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\log \left(e^{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \le 792592107.1486229:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r654698 = x;
        double r654699 = 1.0;
        double r654700 = r654698 + r654699;
        double r654701 = n;
        double r654702 = r654699 / r654701;
        double r654703 = pow(r654700, r654702);
        double r654704 = pow(r654698, r654702);
        double r654705 = r654703 - r654704;
        return r654705;
}

↓

double f(double x, double n) {
        double r654706 = n;
        double r654707 = -1982989920422248.0;
        bool r654708 = r654706 <= r654707;
        double r654709 = 1.0;
        double r654710 = x;
        double r654711 = r654710 * r654706;
        double r654712 = r654709 / r654711;
        double r654713 = log(r654710);
        double r654714 = r654713 / r654710;
        double r654715 = -r654714;
        double r654716 = r654706 * r654706;
        double r654717 = r654715 / r654716;
        double r654718 = r654712 - r654717;
        double r654719 = 0.5;
        double r654720 = r654710 * r654710;
        double r654721 = r654706 * r654720;
        double r654722 = r654719 / r654721;
        double r654723 = r654718 - r654722;
        double r654724 = -4.76046101311837e-310;
        bool r654725 = r654706 <= r654724;
        double r654726 = r654709 + r654710;
        double r654727 = r654709 / r654706;
        double r654728 = pow(r654726, r654727);
        double r654729 = pow(r654710, r654727);
        double r654730 = r654728 - r654729;
        double r654731 = cbrt(r654729);
        double r654732 = r654731 * r654731;
        double r654733 = -r654732;
        double r654734 = r654731 * r654733;
        double r654735 = fma(r654709, r654728, r654734);
        double r654736 = exp(r654735);
        double r654737 = log(r654736);
        double r654738 = r654737 * r654730;
        double r654739 = r654730 * r654738;
        double r654740 = cbrt(r654739);
        double r654741 = cbrt(r654740);
        double r654742 = cbrt(r654730);
        double r654743 = r654742 * r654742;
        double r654744 = r654741 * r654743;
        double r654745 = 792592107.1486229;
        bool r654746 = r654706 <= r654745;
        double r654747 = log1p(r654710);
        double r654748 = r654747 / r654706;
        double r654749 = exp(r654748);
        double r654750 = r654749 - r654729;
        double r654751 = r654746 ? r654750 : r654723;
        double r654752 = r654725 ? r654744 : r654751;
        double r654753 = r654708 ? r654723 : r654752;
        return r654753;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -1982989920422248.0 or 792592107.1486229 < n

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.0

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

   3. Simplified 32.0

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

   if -1982989920422248.0 < n < -4.76046101311837e-310

   1. Initial program 1.8

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

   3. Applied add-cube-cbrt 1.8

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

   5. Applied add-cbrt-cube 1.8

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

   7. Applied add-log-exp 1.9

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

   9. Applied add-cube-cbrt 1.9

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

   10. Applied *-un-lft-identity 1.9

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

   11. Applied prod-diff 1.9

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

   12. Simplified 1.9

       \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\left(\log \left(e^{\mathsf{fma}\left(1, {\left(x + 1\right)}^{\left(\frac{1}{n}\right)}, -\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right) + \color{blue}{0}}\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}}\]

   if -4.76046101311837e-310 < n < 792592107.1486229

   1. Initial program 24.1

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

   3. Applied add-exp-log 24.2

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

   4. Applied pow-exp 24.2

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

   5. Simplified 1.8

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

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1982989920422248.0:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;n \le -4.76046101311837 \cdot 10^{-310}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{\left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right) \cdot \left(\log \left(e^{\mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \left(-\sqrt[3]{{x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{x}^{\left(\frac{1}{n}\right)}}\right)\right)}\right) \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right)}} \cdot \left(\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{elif}\;n \le 792592107.1486229:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \frac{-\frac{\log x}{x}}{n \cdot n}\right) - \frac{\frac{1}{2}}{n \cdot \left(x \cdot x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))