2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.4

Time: 25.5s

Precision: 64

• could not determine a ground truth for program body

  1. x = 2.931459133389821e+208
  2. n = 3.625484546781197e-95

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -1111966062034791:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) - \left(-1\right) \cdot \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \le 251407.4974749986722599714994430541992188:\\ \;\;\;\;\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{x \cdot n} - \frac{-\frac{\log x}{x}}{n} \cdot \frac{0.25}{n}\right) - \log \left(e^{\frac{0.25}{x \cdot \left(x \cdot n\right)}}\right)\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r54446 = x;
        double r54447 = 1.0;
        double r54448 = r54446 + r54447;
        double r54449 = n;
        double r54450 = r54447 / r54449;
        double r54451 = pow(r54448, r54450);
        double r54452 = pow(r54446, r54450);
        double r54453 = r54451 - r54452;
        return r54453;
}

↓

double f(double x, double n) {
        double r54454 = n;
        double r54455 = -1111966062034791.0;
        bool r54456 = r54454 <= r54455;
        double r54457 = 1.0;
        double r54458 = r54457 / r54454;
        double r54459 = x;
        double r54460 = r54458 / r54459;
        double r54461 = 0.5;
        double r54462 = r54459 * r54459;
        double r54463 = r54454 * r54462;
        double r54464 = r54461 / r54463;
        double r54465 = r54460 - r54464;
        double r54466 = -r54457;
        double r54467 = log(r54459);
        double r54468 = r54454 * r54454;
        double r54469 = r54459 * r54468;
        double r54470 = r54467 / r54469;
        double r54471 = r54466 * r54470;
        double r54472 = r54465 - r54471;
        double r54473 = 251407.49747499867;
        bool r54474 = r54454 <= r54473;
        double r54475 = r54457 + r54459;
        double r54476 = pow(r54475, r54458);
        double r54477 = sqrt(r54476);
        double r54478 = pow(r54459, r54458);
        double r54479 = sqrt(r54478);
        double r54480 = r54477 - r54479;
        double r54481 = r54477 + r54479;
        double r54482 = r54480 * r54481;
        double r54483 = r54459 * r54454;
        double r54484 = r54461 / r54483;
        double r54485 = r54467 / r54459;
        double r54486 = -r54485;
        double r54487 = r54486 / r54454;
        double r54488 = 0.25;
        double r54489 = r54488 / r54454;
        double r54490 = r54487 * r54489;
        double r54491 = r54484 - r54490;
        double r54492 = r54459 * r54483;
        double r54493 = r54488 / r54492;
        double r54494 = exp(r54493);
        double r54495 = log(r54494);
        double r54496 = r54491 - r54495;
        double r54497 = r54496 * r54481;
        double r54498 = r54474 ? r54482 : r54497;
        double r54499 = r54456 ? r54472 : r54498;
        return r54499;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if n < -1111966062034791.0

   1. Initial program 44.2

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

   2. Simplified 44.2

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

   4. Applied add-sqr-sqrt 44.2

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

   5. Taylor expanded around inf 31.6

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

   6. Simplified 31.1

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

   if -1111966062034791.0 < n < 251407.49747499867

   1. Initial program 9.1

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

   2. Simplified 9.1

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

   4. Applied add-sqr-sqrt 9.2

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

   5. Applied add-sqr-sqrt 9.1

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

   6. Applied difference-of-squares 9.1

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

   7. Simplified 9.1

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

   8. Simplified 9.1

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

   if 251407.49747499867 < n

   1. Initial program 43.9

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

   2. Simplified 43.9

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

   4. Applied add-sqr-sqrt 44.0

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

   5. Applied add-sqr-sqrt 44.0

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

   6. Applied difference-of-squares 43.9

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

   7. Simplified 43.9

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

   8. Simplified 43.9

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

   9. Taylor expanded around inf 33.0

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

   10. Simplified 33.0

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

   12. Applied add-log-exp 33.0

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

   13. Simplified 33.0

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

4. Final simplification 22.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -1111966062034791:\\ \;\;\;\;\left(\frac{\frac{1}{n}}{x} - \frac{0.5}{n \cdot \left(x \cdot x\right)}\right) - \left(-1\right) \cdot \frac{\log x}{x \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;n \le 251407.4974749986722599714994430541992188:\\ \;\;\;\;\left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{0.5}{x \cdot n} - \frac{-\frac{\log x}{x}}{n} \cdot \frac{0.25}{n}\right) - \log \left(e^{\frac{0.25}{x \cdot \left(x \cdot n\right)}}\right)\right) \cdot \left(\sqrt{{\left(1 + x\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{{x}^{\left(\frac{1}{n}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))