2nthrt (problem 3.4.6)

Average Error: 28.9 → 18.8

Time: 47.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 7.015171298646812e+194
  2. n = 4.52205504228614e-15

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -29869477.165418986:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\log x}{n}}{x \cdot n}\right) - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{elif}\;n \le -3.1374879532667245 \cdot 10^{-302}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;n \le 0.9292141684675859:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n}}{x \cdot n} - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\right) + \frac{\frac{1}{n}}{x}\\ \end{array}\]

C

double f(double x, double n) {
        double r2550409 = x;
        double r2550410 = 1.0;
        double r2550411 = r2550409 + r2550410;
        double r2550412 = n;
        double r2550413 = r2550410 / r2550412;
        double r2550414 = pow(r2550411, r2550413);
        double r2550415 = pow(r2550409, r2550413);
        double r2550416 = r2550414 - r2550415;
        return r2550416;
}

↓

double f(double x, double n) {
        double r2550417 = n;
        double r2550418 = -29869477.165418986;
        bool r2550419 = r2550417 <= r2550418;
        double r2550420 = 1.0;
        double r2550421 = x;
        double r2550422 = r2550421 * r2550417;
        double r2550423 = r2550420 / r2550422;
        double r2550424 = log(r2550421);
        double r2550425 = r2550424 / r2550417;
        double r2550426 = r2550425 / r2550422;
        double r2550427 = r2550423 + r2550426;
        double r2550428 = 0.5;
        double r2550429 = r2550421 * r2550422;
        double r2550430 = r2550428 / r2550429;
        double r2550431 = r2550427 - r2550430;
        double r2550432 = -3.1374879532667245e-302;
        bool r2550433 = r2550417 <= r2550432;
        double r2550434 = r2550420 + r2550421;
        double r2550435 = r2550420 / r2550417;
        double r2550436 = pow(r2550434, r2550435);
        double r2550437 = pow(r2550421, r2550435);
        double r2550438 = r2550436 - r2550437;
        double r2550439 = cbrt(r2550438);
        double r2550440 = r2550439 * r2550439;
        double r2550441 = exp(r2550440);
        double r2550442 = log(r2550441);
        double r2550443 = cbrt(r2550421);
        double r2550444 = pow(r2550443, r2550435);
        double r2550445 = -r2550444;
        double r2550446 = r2550443 * r2550443;
        double r2550447 = pow(r2550446, r2550435);
        double r2550448 = r2550447 * r2550444;
        double r2550449 = fma(r2550445, r2550447, r2550448);
        double r2550450 = -r2550448;
        double r2550451 = fma(r2550420, r2550436, r2550450);
        double r2550452 = r2550449 + r2550451;
        double r2550453 = cbrt(r2550452);
        double r2550454 = r2550442 * r2550453;
        double r2550455 = 0.9292141684675859;
        bool r2550456 = r2550417 <= r2550455;
        double r2550457 = log1p(r2550421);
        double r2550458 = r2550457 / r2550417;
        double r2550459 = exp(r2550458);
        double r2550460 = r2550459 - r2550437;
        double r2550461 = exp(r2550460);
        double r2550462 = log(r2550461);
        double r2550463 = r2550426 - r2550430;
        double r2550464 = r2550435 / r2550421;
        double r2550465 = r2550463 + r2550464;
        double r2550466 = r2550456 ? r2550462 : r2550465;
        double r2550467 = r2550433 ? r2550454 : r2550466;
        double r2550468 = r2550419 ? r2550431 : r2550467;
        return r2550468;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if n < -29869477.165418986

   1. Initial program 45.3

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

   2. Taylor expanded around inf 32.7

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

   3. Simplified 32.7

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

   if -29869477.165418986 < n < -3.1374879532667245e-302

   1. Initial program 0.8

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

   3. Applied add-log-exp 1.1

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

   4. Applied add-log-exp 1.0

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

   5. Applied diff-log 1.0

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

   6. Simplified 1.0

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

   8. Applied add-cube-cbrt 1.0

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

   9. Applied exp-prod 1.0

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

   10. Applied log-pow 1.0

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

   12. Applied add-cube-cbrt 1.0

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

   13. Applied unpow-prod-down 1.0

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

   14. Applied *-un-lft-identity 1.0

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

   15. Applied prod-diff 1.0

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

   if -3.1374879532667245e-302 < n < 0.9292141684675859

   1. Initial program 22.6

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

   3. Applied add-log-exp 22.7

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

   4. Applied add-log-exp 22.7

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

   5. Applied diff-log 22.7

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

   6. Simplified 22.7

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

   8. Applied add-exp-log 23.9

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

   9. Applied pow-exp 23.9

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

   10. Simplified 2.0

       \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)\]

   if 0.9292141684675859 < n

   1. Initial program 43.4

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

   3. Applied add-log-exp 43.5

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

   4. Applied add-log-exp 43.5

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

   5. Applied diff-log 43.5

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

   6. Simplified 43.5

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

   7. Taylor expanded around inf 31.5

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

   8. Simplified 30.8

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} + \left(\frac{\frac{\log x}{n}}{n \cdot x} - \frac{\frac{1}{2}}{\left(n \cdot x\right) \cdot x}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -29869477.165418986:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\frac{\log x}{n}}{x \cdot n}\right) - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\\ \mathbf{elif}\;n \le -3.1374879532667245 \cdot 10^{-302}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) \cdot \sqrt[3]{\mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(1, {\left(1 + x\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)}\\ \mathbf{elif}\;n \le 0.9292141684675859:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\log x}{n}}{x \cdot n} - \frac{\frac{1}{2}}{x \cdot \left(x \cdot n\right)}\right) + \frac{\frac{1}{n}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))