2nthrt (problem 3.4.6)

Average Error: 28.8 → 21.5

Time: 11.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 5.946136944105139e+299
  2. n = 1.5326699225967172e-86

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -7365720.8177897743880748748779296875:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{elif}\;n \le 100.3346359568117804883513599634170532227:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r84598 = x;
        double r84599 = 1.0;
        double r84600 = r84598 + r84599;
        double r84601 = n;
        double r84602 = r84599 / r84601;
        double r84603 = pow(r84600, r84602);
        double r84604 = pow(r84598, r84602);
        double r84605 = r84603 - r84604;
        return r84605;
}

↓

double f(double x, double n) {
        double r84606 = n;
        double r84607 = -7365720.817789774;
        bool r84608 = r84606 <= r84607;
        double r84609 = x;
        double r84610 = 1.0;
        double r84611 = r84609 + r84610;
        double r84612 = r84610 / r84606;
        double r84613 = pow(r84611, r84612);
        double r84614 = sqrt(r84613);
        double r84615 = 2.0;
        double r84616 = r84612 / r84615;
        double r84617 = pow(r84609, r84616);
        double r84618 = r84614 + r84617;
        double r84619 = 0.25;
        double r84620 = 1.0;
        double r84621 = pow(r84609, r84615);
        double r84622 = r84621 * r84606;
        double r84623 = exp(r84622);
        double r84624 = log(r84623);
        double r84625 = r84620 / r84624;
        double r84626 = r84620 / r84609;
        double r84627 = log(r84626);
        double r84628 = pow(r84606, r84615);
        double r84629 = r84609 * r84628;
        double r84630 = r84627 / r84629;
        double r84631 = r84625 + r84630;
        double r84632 = r84619 * r84631;
        double r84633 = -r84632;
        double r84634 = 0.5;
        double r84635 = r84634 / r84606;
        double r84636 = r84635 / r84609;
        double r84637 = r84633 + r84636;
        double r84638 = r84618 * r84637;
        double r84639 = 100.33463595681178;
        bool r84640 = r84606 <= r84639;
        double r84641 = r84614 - r84617;
        double r84642 = r84618 * r84641;
        double r84643 = r84620 / r84622;
        double r84644 = log(r84629);
        double r84645 = exp(r84644);
        double r84646 = r84627 / r84645;
        double r84647 = r84643 + r84646;
        double r84648 = r84619 * r84647;
        double r84649 = -r84648;
        double r84650 = r84649 + r84636;
        double r84651 = r84618 * r84650;
        double r84652 = r84640 ? r84642 : r84651;
        double r84653 = r84608 ? r84638 : r84652;
        return r84653;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -7365720.817789774

   1. Initial program 44.7

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

   3. Applied sqr-pow 44.7

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

   4. Applied add-sqr-sqrt 44.7

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

   5. Applied difference-of-squares 44.7

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

   6. Taylor expanded around inf 31.8

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

   7. Simplified 31.3

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)}\]
   8. Using strategy rm

   9. Applied add-log-exp 31.4

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]

   if -7365720.817789774 < n < 100.33463595681178

   1. Initial program 7.9

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

   3. Applied sqr-pow 8.0

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

   4. Applied add-sqr-sqrt 7.9

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

   5. Applied difference-of-squares 7.9

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

   if 100.33463595681178 < n

   1. Initial program 43.9

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

   3. Applied sqr-pow 44.0

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

   4. Applied add-sqr-sqrt 43.9

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

   5. Applied difference-of-squares 43.9

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

   6. Taylor expanded around inf 32.4

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(0.5 \cdot \frac{1}{x \cdot n} - \left(0.25 \cdot \frac{1}{{x}^{2} \cdot n} + 0.25 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right)}\]

   7. Simplified 31.9

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \color{blue}{\left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)}\]
   8. Using strategy rm

   9. Applied add-exp-log 31.9

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {\color{blue}{\left(e^{\log n}\right)}}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]

   10. Applied pow-exp 31.9

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot \color{blue}{e^{\log n \cdot 2}}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]

   11. Applied add-exp-log 31.9

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{\color{blue}{e^{\log x}} \cdot e^{\log n \cdot 2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]

   12. Applied prod-exp 31.9

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{\color{blue}{e^{\log x + \log n \cdot 2}}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]

   13. Simplified 31.9

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{e^{\color{blue}{\log \left(x \cdot {n}^{2}\right)}}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 21.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -7365720.8177897743880748748779296875:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \mathbf{elif}\;n \le 100.3346359568117804883513599634170532227:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\left(-0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{\log \left(\frac{1}{x}\right)}{e^{\log \left(x \cdot {n}^{2}\right)}}\right)\right) + \frac{\frac{0.5}{n}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))