2nthrt (problem 3.4.6)

Average Error: 28.8 → 22.2

Time: 26.1s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.3180249475760962e+302
  2. n = 1.0692647045277341e-165

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -3134953497269160640512:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{-\log x}{e^{e^{\log \left(\log \left(x \cdot {n}^{2}\right)\right)}}}\right)\right)\\ \mathbf{elif}\;n \le 11998314702.362140655517578125:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{-\log x}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r57333 = x;
        double r57334 = 1.0;
        double r57335 = r57333 + r57334;
        double r57336 = n;
        double r57337 = r57334 / r57336;
        double r57338 = pow(r57335, r57337);
        double r57339 = pow(r57333, r57337);
        double r57340 = r57338 - r57339;
        return r57340;
}

↓

double f(double x, double n) {
        double r57341 = n;
        double r57342 = -3.1349534972691606e+21;
        bool r57343 = r57341 <= r57342;
        double r57344 = x;
        double r57345 = 1.0;
        double r57346 = r57344 + r57345;
        double r57347 = r57345 / r57341;
        double r57348 = pow(r57346, r57347);
        double r57349 = sqrt(r57348);
        double r57350 = 2.0;
        double r57351 = r57347 / r57350;
        double r57352 = pow(r57344, r57351);
        double r57353 = r57349 + r57352;
        double r57354 = 0.5;
        double r57355 = r57344 * r57341;
        double r57356 = r57354 / r57355;
        double r57357 = 0.25;
        double r57358 = 1.0;
        double r57359 = pow(r57344, r57350);
        double r57360 = r57359 * r57341;
        double r57361 = r57358 / r57360;
        double r57362 = log(r57344);
        double r57363 = -r57362;
        double r57364 = pow(r57341, r57350);
        double r57365 = r57344 * r57364;
        double r57366 = log(r57365);
        double r57367 = log(r57366);
        double r57368 = exp(r57367);
        double r57369 = exp(r57368);
        double r57370 = r57363 / r57369;
        double r57371 = r57361 + r57370;
        double r57372 = r57357 * r57371;
        double r57373 = r57356 - r57372;
        double r57374 = r57353 * r57373;
        double r57375 = 11998314702.36214;
        bool r57376 = r57341 <= r57375;
        double r57377 = r57349 - r57352;
        double r57378 = r57353 * r57377;
        double r57379 = exp(r57360);
        double r57380 = log(r57379);
        double r57381 = r57358 / r57380;
        double r57382 = r57363 / r57365;
        double r57383 = r57381 + r57382;
        double r57384 = r57357 * r57383;
        double r57385 = r57356 - r57384;
        double r57386 = r57353 * r57385;
        double r57387 = r57376 ? r57378 : r57386;
        double r57388 = r57343 ? r57374 : r57387;
        return r57388;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -3.1349534972691606e+21

   1. Initial program 44.0

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
   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 44.0

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

   5. Applied difference-of-squares 44.0

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

   6. Taylor expanded around inf 32.3

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

   7. Simplified 32.3

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

   9. Applied add-exp-log 64.0

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

   10. Applied pow-exp 64.0

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

   11. Applied add-exp-log 64.0

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

   12. Applied prod-exp 64.0

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

   13. Simplified 32.3

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

   15. Applied add-exp-log 32.3

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

   if -3.1349534972691606e+21 < n < 11998314702.36214

   1. Initial program 9.3

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

   3. Applied sqr-pow 9.3

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

   4. Applied add-sqr-sqrt 9.3

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

   5. Applied difference-of-squares 9.3

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

   if 11998314702.36214 < n

   1. Initial program 45.1

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

   3. Applied sqr-pow 45.2

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

   4. Applied add-sqr-sqrt 45.2

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

   5. Applied difference-of-squares 45.2

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

   6. Taylor expanded around inf 32.8

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

   7. Simplified 32.8

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

   9. Applied add-log-exp 32.9

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -3134953497269160640512:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{{x}^{2} \cdot n} + \frac{-\log x}{e^{e^{\log \left(\log \left(x \cdot {n}^{2}\right)\right)}}}\right)\right)\\ \mathbf{elif}\;n \le 11998314702.362140655517578125:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + {x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left(\frac{0.5}{x \cdot n} - 0.25 \cdot \left(\frac{1}{\log \left(e^{{x}^{2} \cdot n}\right)} + \frac{-\log x}{x \cdot {n}^{2}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))