2nthrt (problem 3.4.6)

Average Error: 29.4 → 19.2

Time: 2.5m

Precision: 64

• could not determine a ground truth for program body

  1. x = 6.938076856826331e+91
  2. n = 2.2912373152655157e-67

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -71967423679655.73:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{x}\right), \left(\frac{1}{x \cdot n}\right), \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{\left(x \cdot n\right) \cdot n}\right)\right)\right)\\ \mathbf{elif}\;n \le -2.4130801716763244 \cdot 10^{-303}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) - \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;n \le 215979027.38556987:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{x}\right), \left(\frac{1}{x \cdot n}\right), \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{\left(x \cdot n\right) \cdot n}\right)\right)\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r15627371 = x;
        double r15627372 = 1.0;
        double r15627373 = r15627371 + r15627372;
        double r15627374 = n;
        double r15627375 = r15627372 / r15627374;
        double r15627376 = pow(r15627373, r15627375);
        double r15627377 = pow(r15627371, r15627375);
        double r15627378 = r15627376 - r15627377;
        return r15627378;
}

↓

double f(double x, double n) {
        double r15627379 = n;
        double r15627380 = -71967423679655.73;
        bool r15627381 = r15627379 <= r15627380;
        double r15627382 = -0.5;
        double r15627383 = x;
        double r15627384 = r15627382 / r15627383;
        double r15627385 = 1.0;
        double r15627386 = r15627383 * r15627379;
        double r15627387 = r15627385 / r15627386;
        double r15627388 = log(r15627383);
        double r15627389 = r15627386 * r15627379;
        double r15627390 = r15627388 / r15627389;
        double r15627391 = -r15627390;
        double r15627392 = r15627387 - r15627391;
        double r15627393 = fma(r15627384, r15627387, r15627392);
        double r15627394 = -2.4130801716763244e-303;
        bool r15627395 = r15627379 <= r15627394;
        double r15627396 = r15627385 + r15627383;
        double r15627397 = r15627385 / r15627379;
        double r15627398 = pow(r15627396, r15627397);
        double r15627399 = pow(r15627383, r15627397);
        double r15627400 = r15627398 - r15627399;
        double r15627401 = exp(r15627400);
        double r15627402 = cbrt(r15627401);
        double r15627403 = log(r15627402);
        double r15627404 = exp(r15627398);
        double r15627405 = cbrt(r15627404);
        double r15627406 = r15627402 * r15627405;
        double r15627407 = log(r15627406);
        double r15627408 = exp(r15627399);
        double r15627409 = cbrt(r15627408);
        double r15627410 = log(r15627409);
        double r15627411 = r15627407 - r15627410;
        double r15627412 = r15627403 + r15627411;
        double r15627413 = 215979027.38556987;
        bool r15627414 = r15627379 <= r15627413;
        double r15627415 = log1p(r15627383);
        double r15627416 = r15627415 / r15627379;
        double r15627417 = exp(r15627416);
        double r15627418 = r15627417 - r15627399;
        double r15627419 = exp(r15627418);
        double r15627420 = log(r15627419);
        double r15627421 = r15627414 ? r15627420 : r15627393;
        double r15627422 = r15627395 ? r15627412 : r15627421;
        double r15627423 = r15627381 ? r15627393 : r15627422;
        return r15627423;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -71967423679655.73 or 215979027.38556987 < n

   1. Initial program 45.0

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

   3. Applied add-log-exp 45.0

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

   4. Taylor expanded around inf 32.0

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

   5. Simplified 32.0

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

   if -71967423679655.73 < n < -2.4130801716763244e-303

   1. Initial program 1.9

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

   3. Applied add-log-exp 2.0

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

   5. Applied add-cube-cbrt 2.1

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

   6. Applied log-prod 2.1

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

   8. Applied exp-diff 2.1

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

   9. Applied cbrt-div 2.1

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

   10. Applied associate-*l/ 2.1

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

   11. Applied log-div 2.1

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

   if -2.4130801716763244e-303 < n < 215979027.38556987

   1. Initial program 24.0

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

   3. Applied add-log-exp 24.1

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

   5. Applied add-exp-log 25.3

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

   6. Applied pow-exp 25.3

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

   7. Simplified 3.2

      \[\leadsto \log \left(e^{e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 19.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -71967423679655.73:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{x}\right), \left(\frac{1}{x \cdot n}\right), \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{\left(x \cdot n\right) \cdot n}\right)\right)\right)\\ \mathbf{elif}\;n \le -2.4130801716763244 \cdot 10^{-303}:\\ \;\;\;\;\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}}\right) + \left(\log \left(\sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}} \cdot \sqrt[3]{e^{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}}}\right) - \log \left(\sqrt[3]{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)\right)\\ \mathbf{elif}\;n \le 215979027.38556987:\\ \;\;\;\;\log \left(e^{e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\frac{-1}{2}}{x}\right), \left(\frac{1}{x \cdot n}\right), \left(\frac{1}{x \cdot n} - \left(-\frac{\log x}{\left(x \cdot n\right) \cdot n}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))