2nthrt (problem 3.4.6)

Average Error: 29.4 → 19.4

Time: 40.3s

Precision: 64

• could not determine a ground truth for program body

  1. x = 4.035271246258652e+292
  2. n = 2.011286021475988e-116

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;n \le -155965176.56480727:\\ \;\;\;\;\left(\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{1}{2}}{x \cdot n} - \mathsf{fma}\left(\left(-\frac{\frac{\log x}{x}}{n \cdot n}\right), \frac{1}{4}, \left(\frac{\frac{1}{4}}{\left(x \cdot n\right) \cdot x}\right)\right)\right)\\ \mathbf{elif}\;n \le -2.5874953812279278 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right)} \cdot \left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right)\right)\\ \mathbf{elif}\;n \le 722234431.8548901:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{n} - \frac{\log x}{\left(n \cdot n\right) \cdot x}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r1964558 = x;
        double r1964559 = 1.0;
        double r1964560 = r1964558 + r1964559;
        double r1964561 = n;
        double r1964562 = r1964559 / r1964561;
        double r1964563 = pow(r1964560, r1964562);
        double r1964564 = pow(r1964558, r1964562);
        double r1964565 = r1964563 - r1964564;
        return r1964565;
}

↓

double f(double x, double n) {
        double r1964566 = n;
        double r1964567 = -155965176.56480727;
        bool r1964568 = r1964566 <= r1964567;
        double r1964569 = x;
        double r1964570 = 1.0;
        double r1964571 = r1964570 / r1964566;
        double r1964572 = pow(r1964569, r1964571);
        double r1964573 = log1p(r1964572);
        double r1964574 = expm1(r1964573);
        double r1964575 = sqrt(r1964574);
        double r1964576 = r1964569 + r1964570;
        double r1964577 = pow(r1964576, r1964571);
        double r1964578 = sqrt(r1964577);
        double r1964579 = r1964575 + r1964578;
        double r1964580 = 0.5;
        double r1964581 = r1964569 * r1964566;
        double r1964582 = r1964580 / r1964581;
        double r1964583 = log(r1964569);
        double r1964584 = r1964583 / r1964569;
        double r1964585 = r1964566 * r1964566;
        double r1964586 = r1964584 / r1964585;
        double r1964587 = -r1964586;
        double r1964588 = 0.25;
        double r1964589 = r1964581 * r1964569;
        double r1964590 = r1964588 / r1964589;
        double r1964591 = fma(r1964587, r1964588, r1964590);
        double r1964592 = r1964582 - r1964591;
        double r1964593 = r1964579 * r1964592;
        double r1964594 = -2.5874953812279278e-300;
        bool r1964595 = r1964566 <= r1964594;
        double r1964596 = r1964578 - r1964575;
        double r1964597 = cbrt(r1964596);
        double r1964598 = r1964597 * r1964597;
        double r1964599 = r1964597 * r1964598;
        double r1964600 = cbrt(r1964599);
        double r1964601 = r1964600 * r1964598;
        double r1964602 = r1964579 * r1964601;
        double r1964603 = 722234431.8548901;
        bool r1964604 = r1964566 <= r1964603;
        double r1964605 = log1p(r1964569);
        double r1964606 = r1964605 * r1964571;
        double r1964607 = exp(r1964606);
        double r1964608 = r1964607 - r1964572;
        double r1964609 = r1964571 / r1964569;
        double r1964610 = r1964580 / r1964569;
        double r1964611 = r1964610 / r1964569;
        double r1964612 = r1964611 / r1964566;
        double r1964613 = r1964585 * r1964569;
        double r1964614 = r1964583 / r1964613;
        double r1964615 = r1964612 - r1964614;
        double r1964616 = r1964609 - r1964615;
        double r1964617 = r1964604 ? r1964608 : r1964616;
        double r1964618 = r1964595 ? r1964602 : r1964617;
        double r1964619 = r1964568 ? r1964593 : r1964618;
        return r1964619;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 4 regimes

2. if n < -155965176.56480727

   1. Initial program 44.5

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

   3. Applied expm1-log1p-u 44.5

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 44.6

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\]

   6. Applied add-sqr-sqrt 44.6

      \[\leadsto \color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]

   7. Applied difference-of-squares 44.6

      \[\leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right)}\]

   8. Taylor expanded around inf 32.8

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{x \cdot n} - \left(\frac{1}{4} \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}} + \frac{1}{4} \cdot \frac{1}{{x}^{2} \cdot n}\right)\right)}\]

   9. Simplified 32.7

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{n \cdot x} - \mathsf{fma}\left(\left(\frac{\frac{-\log x}{x}}{n \cdot n}\right), \frac{1}{4}, \left(\frac{\frac{1}{4}}{\left(n \cdot x\right) \cdot x}\right)\right)\right)}\]

   if -155965176.56480727 < n < -2.5874953812279278e-300

   1. Initial program 0.9

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

   3. Applied expm1-log1p-u 0.9

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.9

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\]

   6. Applied add-sqr-sqrt 0.9

      \[\leadsto \color{blue}{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]

   7. Applied difference-of-squares 0.9

      \[\leadsto \color{blue}{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 0.9

      \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 0.9

       \[\leadsto \left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} + \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\right) \cdot \left(\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right) \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}}}\right)\]

   if -2.5874953812279278e-300 < n < 722234431.8548901

   1. Initial program 21.9

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

   3. Applied add-exp-log 21.9

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

   4. Simplified 2.6

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

   if 722234431.8548901 < 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 expm1-log1p-u 45.1

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}\]

   4. Taylor expanded around inf 32.6

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

   5. Simplified 32.1

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

4. Final simplification 19.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \le -155965176.56480727:\\ \;\;\;\;\left(\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\frac{\frac{1}{2}}{x \cdot n} - \mathsf{fma}\left(\left(-\frac{\frac{\log x}{x}}{n \cdot n}\right), \frac{1}{4}, \left(\frac{\frac{1}{4}}{\left(x \cdot n\right) \cdot x}\right)\right)\right)\\ \mathbf{elif}\;n \le -2.5874953812279278 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)} + \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right)} \cdot \left(\sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}} \cdot \sqrt[3]{\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}} - \sqrt{\mathsf{expm1}\left(\left(\mathsf{log1p}\left(\left({x}^{\left(\frac{1}{n}\right)}\right)\right)\right)\right)}}\right)\right)\\ \mathbf{elif}\;n \le 722234431.8548901:\\ \;\;\;\;e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{\frac{\frac{1}{2}}{x}}{x}}{n} - \frac{\log x}{\left(n \cdot n\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))