2nthrt (problem 3.4.6)

Average Error: 29.9 → 23.1

Time: 30.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 7.953652505345186e+78
  2. n = 2.911049425072326e-200

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.573264323221321940469903236644308698369 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.331213512681894439398036102681617494142 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \log \left(e^{\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r71390 = x;
        double r71391 = 1.0;
        double r71392 = r71390 + r71391;
        double r71393 = n;
        double r71394 = r71391 / r71393;
        double r71395 = pow(r71392, r71394);
        double r71396 = pow(r71390, r71394);
        double r71397 = r71395 - r71396;
        return r71397;
}

↓

double f(double x, double n) {
        double r71398 = 1.0;
        double r71399 = n;
        double r71400 = r71398 / r71399;
        double r71401 = -1.573264323221322e-23;
        bool r71402 = r71400 <= r71401;
        double r71403 = x;
        double r71404 = r71403 + r71398;
        double r71405 = 2.0;
        double r71406 = r71400 / r71405;
        double r71407 = pow(r71404, r71406);
        double r71408 = pow(r71403, r71406);
        double r71409 = r71407 + r71408;
        double r71410 = r71407 - r71408;
        double r71411 = exp(r71410);
        double r71412 = cbrt(r71411);
        double r71413 = log(r71412);
        double r71414 = r71405 * r71413;
        double r71415 = r71414 + r71413;
        double r71416 = r71409 * r71415;
        double r71417 = 2.3312135126818944e-13;
        bool r71418 = r71400 <= r71417;
        double r71419 = r71398 / r71403;
        double r71420 = 1.0;
        double r71421 = r71420 / r71399;
        double r71422 = log(r71403);
        double r71423 = -r71422;
        double r71424 = pow(r71399, r71405);
        double r71425 = r71423 / r71424;
        double r71426 = r71421 - r71425;
        double r71427 = r71419 * r71426;
        double r71428 = 0.5;
        double r71429 = pow(r71403, r71405);
        double r71430 = r71429 * r71399;
        double r71431 = r71428 / r71430;
        double r71432 = r71427 - r71431;
        double r71433 = pow(r71404, r71400);
        double r71434 = pow(r71403, r71400);
        double r71435 = r71433 - r71434;
        double r71436 = r71408 + r71407;
        double r71437 = r71435 / r71436;
        double r71438 = exp(r71437);
        double r71439 = log(r71438);
        double r71440 = r71409 * r71439;
        double r71441 = r71418 ? r71432 : r71440;
        double r71442 = r71402 ? r71416 : r71441;
        return r71442;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -1.573264323221322e-23

   1. Initial program 3.5

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

   3. Applied sqr-pow 3.6

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

   4. Applied sqr-pow 3.5

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

   5. Applied difference-of-squares 3.5

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

   7. Applied add-log-exp 3.7

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

   8. Applied add-log-exp 3.6

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

   9. Applied diff-log 3.7

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

   10. Simplified 3.6

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

   12. Applied add-cube-cbrt 3.7

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

   13. Applied log-prod 3.7

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

   14. Simplified 3.7

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

   if -1.573264323221322e-23 < (/ 1.0 n) < 2.3312135126818944e-13

   1. Initial program 44.8

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 31.9

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

   if 2.3312135126818944e-13 < (/ 1.0 n)

   1. Initial program 25.2

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

   3. Applied sqr-pow 25.3

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

   4. Applied sqr-pow 25.2

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

   5. Applied difference-of-squares 25.2

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

   7. Applied add-log-exp 25.3

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

   8. Applied add-log-exp 25.3

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

   9. Applied diff-log 25.3

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

   10. Simplified 25.3

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

   12. Applied flip-- 28.0

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

   13. Simplified 28.0

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

   14. Simplified 28.0

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

4. Final simplification 23.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -1.573264323221321940469903236644308698369 \cdot 10^{-23}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right) + \log \left(\sqrt[3]{e^{{\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\right)\\ \mathbf{elif}\;\frac{1}{n} \le 2.331213512681894439398036102681617494142 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{-\log x}{{n}^{2}}\right) - \frac{0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \log \left(e^{\frac{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}{{x}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(x + 1\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))