2nthrt (problem 3.4.6)

Average Error: 32.5 → 24.0

Time: 11.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.2081374565579042e+172
  2. n = 1.497795285083943e-173

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.032379292376500446:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{e^{\log \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.50994306778113202 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r47310 = x;
        double r47311 = 1.0;
        double r47312 = r47310 + r47311;
        double r47313 = n;
        double r47314 = r47311 / r47313;
        double r47315 = pow(r47312, r47314);
        double r47316 = pow(r47310, r47314);
        double r47317 = r47315 - r47316;
        return r47317;
}

↓

double f(double x, double n) {
        double r47318 = 1.0;
        double r47319 = n;
        double r47320 = r47318 / r47319;
        double r47321 = -0.032379292376500446;
        bool r47322 = r47320 <= r47321;
        double r47323 = x;
        double r47324 = r47323 + r47318;
        double r47325 = pow(r47324, r47320);
        double r47326 = pow(r47323, r47320);
        double r47327 = 3.0;
        double r47328 = pow(r47326, r47327);
        double r47329 = log(r47328);
        double r47330 = exp(r47329);
        double r47331 = cbrt(r47330);
        double r47332 = r47325 - r47331;
        double r47333 = 1.509943067781132e-13;
        bool r47334 = r47320 <= r47333;
        double r47335 = r47320 / r47323;
        double r47336 = 0.5;
        double r47337 = 2.0;
        double r47338 = pow(r47323, r47337);
        double r47339 = r47338 * r47319;
        double r47340 = r47336 / r47339;
        double r47341 = exp(r47340);
        double r47342 = log(r47341);
        double r47343 = log(r47323);
        double r47344 = r47343 * r47318;
        double r47345 = pow(r47319, r47337);
        double r47346 = r47323 * r47345;
        double r47347 = r47344 / r47346;
        double r47348 = r47342 - r47347;
        double r47349 = r47335 - r47348;
        double r47350 = r47337 * r47320;
        double r47351 = pow(r47324, r47350);
        double r47352 = pow(r47323, r47350);
        double r47353 = r47351 - r47352;
        double r47354 = r47325 + r47326;
        double r47355 = r47353 / r47354;
        double r47356 = r47334 ? r47349 : r47355;
        double r47357 = r47322 ? r47332 : r47356;
        return r47357;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.032379292376500446

   1. Initial program 0.2

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

   3. Applied add-cbrt-cube 0.3

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

   4. Simplified 0.3

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

   6. Applied add-exp-log 0.3

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

   7. Applied pow-exp 0.3

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

   8. Applied pow-exp 0.3

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

   9. Simplified 0.3

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

   if -0.032379292376500446 < (/ 1.0 n) < 1.509943067781132e-13

   1. Initial program 44.5

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

   2. Taylor expanded around inf 32.8

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

   3. Simplified 32.3

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

   5. Applied add-log-exp 32.4

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

   6. Simplified 32.4

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

   if 1.509943067781132e-13 < (/ 1.0 n)

   1. Initial program 7.6

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

   3. Applied add-cbrt-cube 7.7

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

   4. Simplified 7.7

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

   6. Applied add-exp-log 7.7

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

   7. Applied pow-exp 7.7

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

   8. Applied pow-exp 7.7

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

   9. Simplified 7.7

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

   11. Applied flip-- 7.8

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

   12. Simplified 7.6

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

   13. Simplified 7.6

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

4. Final simplification 24.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.032379292376500446:\\ \;\;\;\;{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \sqrt[3]{e^{\log \left({\left({x}^{\left(\frac{1}{n}\right)}\right)}^{3}\right)}}\\ \mathbf{elif}\;\frac{1}{n} \le 1.50994306778113202 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\log \left(e^{\frac{0.5}{{x}^{2} \cdot n}}\right) - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))