2nthrt (problem 3.4.6)

Average Error: 29.4 → 22.3

Time: 20.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• could not determine a ground truth for program body

  1. x = 4.773292077298795e+235
  2. n = 2.4612524637435313e-181

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \end{array}\]

C

double f(double x, double n) {
        double r71538 = x;
        double r71539 = 1.0;
        double r71540 = r71538 + r71539;
        double r71541 = n;
        double r71542 = r71539 / r71541;
        double r71543 = pow(r71540, r71542);
        double r71544 = pow(r71538, r71542);
        double r71545 = r71543 - r71544;
        return r71545;
}

↓

double f(double x, double n) {
        double r71546 = 1.0;
        double r71547 = n;
        double r71548 = r71546 / r71547;
        double r71549 = -2.429908229671821e-07;
        bool r71550 = r71548 <= r71549;
        double r71551 = 1.4807923677373035e-15;
        bool r71552 = r71548 <= r71551;
        double r71553 = !r71552;
        bool r71554 = r71550 || r71553;
        double r71555 = x;
        double r71556 = r71555 + r71546;
        double r71557 = pow(r71556, r71548);
        double r71558 = pow(r71555, r71548);
        double r71559 = r71557 - r71558;
        double r71560 = 3.0;
        double r71561 = pow(r71559, r71560);
        double r71562 = cbrt(r71561);
        double r71563 = pow(r71562, r71560);
        double r71564 = cbrt(r71563);
        double r71565 = pow(r71564, r71560);
        double r71566 = cbrt(r71565);
        double r71567 = pow(r71566, r71560);
        double r71568 = cbrt(r71567);
        double r71569 = r71555 * r71547;
        double r71570 = r71546 / r71569;
        double r71571 = log(r71555);
        double r71572 = -r71571;
        double r71573 = 2.0;
        double r71574 = pow(r71547, r71573);
        double r71575 = r71555 * r71574;
        double r71576 = r71572 / r71575;
        double r71577 = 0.5;
        double r71578 = pow(r71555, r71573);
        double r71579 = r71578 * r71547;
        double r71580 = r71577 / r71579;
        double r71581 = fma(r71576, r71546, r71580);
        double r71582 = r71570 - r71581;
        double r71583 = r71554 ? r71568 : r71582;
        return r71583;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 n) < -2.429908229671821e-07 or 1.4807923677373035e-15 < (/ 1.0 n)

   1. Initial program 9.4

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

   3. Applied add-cbrt-cube 9.4

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

   4. Simplified 9.4

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

   6. Applied add-cbrt-cube 9.4

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

   7. Simplified 9.4

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

   9. Applied add-cbrt-cube 9.4

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

   10. Simplified 9.4

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

   12. Applied add-cbrt-cube 9.4

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

   13. Simplified 9.4

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

   if -2.429908229671821e-07 < (/ 1.0 n) < 1.4807923677373035e-15

   1. Initial program 44.9

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

   3. Applied add-cbrt-cube 44.9

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

   4. Simplified 44.9

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

   5. Taylor expanded around inf 32.2

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

   6. Simplified 32.2

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

4. Final simplification 22.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.42990822967182081 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 1.48079236773730348 \cdot 10^{-15}\right):\\ \;\;\;\;\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)}^{3}}\right)}^{3}}\right)}^{3}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot n} - \mathsf{fma}\left(\frac{-\log x}{x \cdot {n}^{2}}, 1, \frac{0.5}{{x}^{2} \cdot n}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))