2nthrt (problem 3.4.6)

Average Error: 29.0 → 22.2

Time: 1.6m

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.239867399397548e+149
  2. n = 1.3192626187298042e-294

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.9753751339402973 \cdot 10^{-18}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.458558234220834 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\right) + \frac{\frac{-1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r12713689 = x;
        double r12713690 = 1.0;
        double r12713691 = r12713689 + r12713690;
        double r12713692 = n;
        double r12713693 = r12713690 / r12713692;
        double r12713694 = pow(r12713691, r12713693);
        double r12713695 = pow(r12713689, r12713693);
        double r12713696 = r12713694 - r12713695;
        return r12713696;
}

↓

double f(double x, double n) {
        double r12713697 = 1.0;
        double r12713698 = n;
        double r12713699 = r12713697 / r12713698;
        double r12713700 = -1.9753751339402973e-18;
        bool r12713701 = r12713699 <= r12713700;
        double r12713702 = x;
        double r12713703 = r12713702 + r12713697;
        double r12713704 = pow(r12713703, r12713699);
        double r12713705 = pow(r12713702, r12713699);
        double r12713706 = r12713704 - r12713705;
        double r12713707 = cbrt(r12713706);
        double r12713708 = r12713707 * r12713707;
        double r12713709 = exp(r12713708);
        double r12713710 = cbrt(r12713709);
        double r12713711 = r12713710 * r12713710;
        double r12713712 = log(r12713711);
        double r12713713 = log(r12713710);
        double r12713714 = r12713712 + r12713713;
        double r12713715 = r12713707 * r12713714;
        double r12713716 = 1.458558234220834e-11;
        bool r12713717 = r12713699 <= r12713716;
        double r12713718 = r12713702 * r12713698;
        double r12713719 = r12713697 / r12713718;
        double r12713720 = log(r12713702);
        double r12713721 = r12713698 * r12713718;
        double r12713722 = r12713720 / r12713721;
        double r12713723 = -r12713722;
        double r12713724 = r12713719 - r12713723;
        double r12713725 = -0.5;
        double r12713726 = r12713718 * r12713702;
        double r12713727 = r12713725 / r12713726;
        double r12713728 = r12713724 + r12713727;
        double r12713729 = exp(r12713707);
        double r12713730 = log(r12713729);
        double r12713731 = r12713730 * r12713707;
        double r12713732 = r12713731 * r12713707;
        double r12713733 = r12713717 ? r12713728 : r12713732;
        double r12713734 = r12713701 ? r12713715 : r12713733;
        return r12713734;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -1.9753751339402973e-18

   1. Initial program 2.1

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

   3. Applied add-log-exp 2.2

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

   5. Applied add-cube-cbrt 2.2

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

   6. Applied exp-prod 2.2

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

   7. Applied log-pow 2.2

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

   9. Applied add-cube-cbrt 2.2

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

   10. Applied log-prod 2.2

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

   if -1.9753751339402973e-18 < (/ 1 n) < 1.458558234220834e-11

   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 32.5

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

   3. Simplified 32.5

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

   if 1.458558234220834e-11 < (/ 1 n)

   1. Initial program 23.8

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

   3. Applied add-log-exp 23.8

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

   5. Applied add-cube-cbrt 23.9

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

   6. Applied exp-prod 23.9

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

   7. Applied log-pow 23.8

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

   9. Applied add-log-exp 23.8

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

   10. Applied exp-to-pow 23.8

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

   11. Applied log-pow 23.8

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.9753751339402973 \cdot 10^{-18}:\\ \;\;\;\;\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \left(\log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}} \cdot \sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.458558234220834 \cdot 10^{-11}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{n \cdot \left(x \cdot n\right)}\right)\right) + \frac{\frac{-1}{2}}{\left(x \cdot n\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(e^{\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))