2nthrt (problem 3.4.6)

Average Error: 29.9 → 23.1

Time: 32.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 7.953652505345186e+78
  2. n = 2.911049425072326e-200

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.573264323221321940469903236644308698369 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.331213512681894439398036102681617494142 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r59719 = x;
        double r59720 = 1.0;
        double r59721 = r59719 + r59720;
        double r59722 = n;
        double r59723 = r59720 / r59722;
        double r59724 = pow(r59721, r59723);
        double r59725 = pow(r59719, r59723);
        double r59726 = r59724 - r59725;
        return r59726;
}

↓

double f(double x, double n) {
        double r59727 = 1.0;
        double r59728 = n;
        double r59729 = r59727 / r59728;
        double r59730 = -1.573264323221322e-23;
        bool r59731 = r59729 <= r59730;
        double r59732 = x;
        double r59733 = r59732 + r59727;
        double r59734 = 2.0;
        double r59735 = r59729 / r59734;
        double r59736 = pow(r59733, r59735);
        double r59737 = pow(r59732, r59729);
        double r59738 = sqrt(r59737);
        double r59739 = r59736 + r59738;
        double r59740 = r59736 - r59738;
        double r59741 = exp(r59740);
        double r59742 = cbrt(r59741);
        double r59743 = log(r59742);
        double r59744 = r59734 * r59743;
        double r59745 = r59744 + r59743;
        double r59746 = r59739 * r59745;
        double r59747 = 2.3312135126818944e-13;
        bool r59748 = r59729 <= r59747;
        double r59749 = r59727 / r59732;
        double r59750 = 1.0;
        double r59751 = r59750 / r59728;
        double r59752 = log(r59732);
        double r59753 = -r59752;
        double r59754 = pow(r59728, r59734);
        double r59755 = r59753 / r59754;
        double r59756 = r59751 - r59755;
        double r59757 = r59749 * r59756;
        double r59758 = 0.5;
        double r59759 = pow(r59732, r59734);
        double r59760 = r59759 * r59728;
        double r59761 = r59758 / r59760;
        double r59762 = r59757 - r59761;
        double r59763 = pow(r59733, r59729);
        double r59764 = r59763 - r59737;
        double r59765 = r59764 / r59739;
        double r59766 = exp(r59765);
        double r59767 = log(r59766);
        double r59768 = r59739 * r59767;
        double r59769 = r59748 ? r59762 : r59768;
        double r59770 = r59731 ? r59746 : r59769;
        return r59770;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.573264323221322e-23

   1. Initial program 3.5

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

   3. Applied add-sqr-sqrt 3.6

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

   4. Applied sqr-pow 3.5

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

   5. Applied difference-of-squares 3.5

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

   7. Applied add-log-exp 3.8

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

   8. Applied add-log-exp 3.7

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

   9. Applied diff-log 3.7

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

   10. Simplified 3.7

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

   12. Applied add-cube-cbrt 3.7

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

   13. Applied log-prod 3.7

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

   14. Simplified 3.7

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

   if -1.573264323221322e-23 < (/ 1.0 n) < 2.3312135126818944e-13

   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 32.6

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

   3. Simplified 31.9

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

   if 2.3312135126818944e-13 < (/ 1.0 n)

   1. Initial program 25.2

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

   3. Applied add-sqr-sqrt 25.3

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

   4. Applied sqr-pow 25.2

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

   5. Applied difference-of-squares 25.2

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

   7. Applied add-log-exp 25.3

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

   8. Applied add-log-exp 25.3

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

   9. Applied diff-log 25.3

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

   10. Simplified 25.3

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

   12. Applied flip-- 28.0

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

   13. Simplified 28.0

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

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.573264323221321940469903236644308698369 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.331213512681894439398036102681617494142 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \log \left(e^{\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))