2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.2

Time: 28.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.0354753041264744e+289
  2. n = 1.805639974384451e-278

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.140830927760539067774892796574941200571 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.080776736319869253764442313662561201966 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{1 \cdot \left(-\log x\right)}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r79676 = x;
        double r79677 = 1.0;
        double r79678 = r79676 + r79677;
        double r79679 = n;
        double r79680 = r79677 / r79679;
        double r79681 = pow(r79678, r79680);
        double r79682 = pow(r79676, r79680);
        double r79683 = r79681 - r79682;
        return r79683;
}

↓

double f(double x, double n) {
        double r79684 = 1.0;
        double r79685 = n;
        double r79686 = r79684 / r79685;
        double r79687 = -1.140830927760539e-09;
        bool r79688 = r79686 <= r79687;
        double r79689 = x;
        double r79690 = r79689 + r79684;
        double r79691 = pow(r79690, r79686);
        double r79692 = pow(r79689, r79686);
        double r79693 = r79691 - r79692;
        double r79694 = exp(r79693);
        double r79695 = sqrt(r79694);
        double r79696 = log(r79695);
        double r79697 = r79696 + r79696;
        double r79698 = 1.0807767363198693e-26;
        bool r79699 = r79686 <= r79698;
        double r79700 = r79686 / r79689;
        double r79701 = log(r79689);
        double r79702 = -r79701;
        double r79703 = r79684 * r79702;
        double r79704 = r79685 * r79685;
        double r79705 = r79689 * r79704;
        double r79706 = r79703 / r79705;
        double r79707 = r79700 - r79706;
        double r79708 = 0.5;
        double r79709 = r79708 / r79685;
        double r79710 = 2.0;
        double r79711 = pow(r79689, r79710);
        double r79712 = r79709 / r79711;
        double r79713 = r79707 - r79712;
        double r79714 = log(r79694);
        double r79715 = r79699 ? r79713 : r79714;
        double r79716 = r79688 ? r79697 : r79715;
        return r79716;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.140830927760539e-09

   1. Initial program 0.7

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

   2. Simplified 0.7

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

   4. Applied add-log-exp 0.9

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

   5. Applied add-log-exp 0.9

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

   6. Applied diff-log 0.9

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

   7. Simplified 0.9

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

   9. Applied add-sqr-sqrt 0.9

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

   10. Applied log-prod 0.9

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

   11. Simplified 0.9

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

   12. Simplified 0.9

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

   if -1.140830927760539e-09 < (/ 1.0 n) < 1.0807767363198693e-26

   1. Initial program 44.7

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

   2. Simplified 44.7

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

   4. Applied add-log-exp 44.7

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

   5. Applied add-log-exp 44.7

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

   6. Applied diff-log 44.7

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

   7. Simplified 44.7

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

   8. Taylor expanded around inf 32.4

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

   9. Simplified 31.8

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

   if 1.0807767363198693e-26 < (/ 1.0 n)

   1. Initial program 27.6

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

   2. Simplified 27.6

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

   4. Applied add-log-exp 27.7

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

   5. Applied add-log-exp 27.7

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

   6. Applied diff-log 27.7

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

   7. Simplified 27.7

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.140830927760539067774892796574941200571 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \log \left(\sqrt{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 1.080776736319869253764442313662561201966 \cdot 10^{-26}:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{1 \cdot \left(-\log x\right)}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{0.5}{n}}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))