2nthrt (problem 3.4.6)

Average Error: 33.0 → 24.0

Time: 21.2s

Precision: 64

• could not determine a ground truth for program body

  1. x = 1.199947025871061e+147
  2. n = 1.4433187213430084e-221

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r117553 = x;
        double r117554 = 1.0;
        double r117555 = r117553 + r117554;
        double r117556 = n;
        double r117557 = r117554 / r117556;
        double r117558 = pow(r117555, r117557);
        double r117559 = pow(r117553, r117557);
        double r117560 = r117558 - r117559;
        return r117560;
}

↓

double f(double x, double n) {
        double r117561 = 1.0;
        double r117562 = n;
        double r117563 = r117561 / r117562;
        double r117564 = -7.745650381978236e-30;
        bool r117565 = r117563 <= r117564;
        double r117566 = x;
        double r117567 = r117566 + r117561;
        double r117568 = pow(r117567, r117563);
        double r117569 = pow(r117566, r117563);
        double r117570 = r117568 - r117569;
        double r117571 = cbrt(r117570);
        double r117572 = -r117569;
        double r117573 = r117568 + r117572;
        double r117574 = cbrt(r117573);
        double r117575 = r117571 * r117574;
        double r117576 = sqrt(r117567);
        double r117577 = pow(r117576, r117563);
        double r117578 = sqrt(r117566);
        double r117579 = pow(r117578, r117563);
        double r117580 = r117577 + r117579;
        double r117581 = cbrt(r117580);
        double r117582 = 2.0;
        double r117583 = r117563 / r117582;
        double r117584 = pow(r117576, r117583);
        double r117585 = pow(r117578, r117583);
        double r117586 = r117584 + r117585;
        double r117587 = r117584 - r117585;
        double r117588 = r117586 * r117587;
        double r117589 = cbrt(r117588);
        double r117590 = r117581 * r117589;
        double r117591 = r117575 * r117590;
        double r117592 = 3.6323271639935607e-11;
        bool r117593 = r117563 <= r117592;
        double r117594 = r117561 / r117566;
        double r117595 = 1.0;
        double r117596 = r117595 / r117562;
        double r117597 = r117595 / r117566;
        double r117598 = log(r117597);
        double r117599 = pow(r117562, r117582);
        double r117600 = r117598 / r117599;
        double r117601 = r117596 - r117600;
        double r117602 = r117594 * r117601;
        double r117603 = 0.5;
        double r117604 = -r117603;
        double r117605 = pow(r117566, r117582);
        double r117606 = r117605 * r117562;
        double r117607 = r117604 / r117606;
        double r117608 = r117602 + r117607;
        double r117609 = r117582 * r117563;
        double r117610 = pow(r117567, r117609);
        double r117611 = pow(r117566, r117609);
        double r117612 = -r117611;
        double r117613 = r117610 + r117612;
        double r117614 = r117568 + r117569;
        double r117615 = r117613 / r117614;
        double r117616 = r117593 ? r117608 : r117615;
        double r117617 = r117565 ? r117591 : r117616;
        return r117617;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -7.745650381978236e-30

   1. Initial program 7.6

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

   3. Applied add-cube-cbrt 7.6

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

   5. Applied add-sqr-sqrt 7.6

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

   6. Applied unpow-prod-down 7.6

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

   7. Applied add-sqr-sqrt 7.6

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

   8. Applied unpow-prod-down 7.6

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

   9. Applied difference-of-squares 7.6

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

   10. Applied cbrt-prod 7.6

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

   12. Applied sqr-pow 7.6

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

   13. Applied sqr-pow 7.6

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

   14. Applied difference-of-squares 7.6

       \[\leadsto \left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\color{blue}{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}\right)\]
   15. Using strategy rm

   16. Applied sub-neg 7.6

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

   if -7.745650381978236e-30 < (/ 1.0 n) < 3.6323271639935607e-11

   1. Initial program 44.9

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

   2. Taylor expanded around inf 32.4

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

   3. Simplified 31.8

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

   if 3.6323271639935607e-11 < (/ 1.0 n)

   1. Initial program 7.4

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

   3. Applied flip-- 7.4

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

   4. Simplified 7.3

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

4. Final simplification 24.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -7.7456503819782359 \cdot 10^{-30}:\\ \;\;\;\;\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + \left(-{x}^{\left(\frac{1}{n}\right)}\right)}\right) \cdot \left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{\left(\frac{1}{n}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{\left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} + {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right) \cdot \left({\left(\sqrt{x + 1}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)} - {\left(\sqrt{x}\right)}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.6323271639935607 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{n} - \frac{\log \left(\frac{1}{x}\right)}{{n}^{2}}\right) + \frac{-0.5}{{x}^{2} \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} + \left(-{x}^{\left(2 \cdot \frac{1}{n}\right)}\right)}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))