2nthrt (problem 3.4.6)

Average Error: 29.4 → 19.0

Time: 39.8s

Precision: 64

• could not determine a ground truth for program body

  1. x = 8.351211544149733e+102
  2. n = 4.308249766459817e-192

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{\left(\left(\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

C

double f(double x, double n) {
        double r3620705 = x;
        double r3620706 = 1.0;
        double r3620707 = r3620705 + r3620706;
        double r3620708 = n;
        double r3620709 = r3620706 / r3620708;
        double r3620710 = pow(r3620707, r3620709);
        double r3620711 = pow(r3620705, r3620709);
        double r3620712 = r3620710 - r3620711;
        return r3620712;
}

↓

double f(double x, double n) {
        double r3620713 = 1.0;
        double r3620714 = n;
        double r3620715 = r3620713 / r3620714;
        double r3620716 = -3.147233419817972e-08;
        bool r3620717 = r3620715 <= r3620716;
        double r3620718 = x;
        double r3620719 = r3620718 + r3620713;
        double r3620720 = pow(r3620719, r3620715);
        double r3620721 = sqrt(r3620720);
        double r3620722 = cbrt(r3620718);
        double r3620723 = pow(r3620722, r3620715);
        double r3620724 = r3620722 * r3620722;
        double r3620725 = pow(r3620724, r3620715);
        double r3620726 = r3620723 * r3620725;
        double r3620727 = -r3620726;
        double r3620728 = fma(r3620721, r3620721, r3620727);
        double r3620729 = -r3620723;
        double r3620730 = fma(r3620729, r3620725, r3620726);
        double r3620731 = r3620728 + r3620730;
        double r3620732 = pow(r3620718, r3620715);
        double r3620733 = r3620720 - r3620732;
        double r3620734 = r3620731 * r3620733;
        double r3620735 = cbrt(r3620733);
        double r3620736 = r3620735 * r3620735;
        double r3620737 = r3620736 * r3620735;
        double r3620738 = r3620734 * r3620737;
        double r3620739 = cbrt(r3620738);
        double r3620740 = 2.220885236034127e-08;
        bool r3620741 = r3620715 <= r3620740;
        double r3620742 = r3620718 * r3620714;
        double r3620743 = r3620713 / r3620742;
        double r3620744 = log(r3620718);
        double r3620745 = r3620714 * r3620714;
        double r3620746 = r3620718 * r3620745;
        double r3620747 = r3620744 / r3620746;
        double r3620748 = r3620743 + r3620747;
        double r3620749 = 0.5;
        double r3620750 = r3620749 / r3620714;
        double r3620751 = r3620718 * r3620718;
        double r3620752 = r3620750 / r3620751;
        double r3620753 = r3620748 - r3620752;
        double r3620754 = log1p(r3620718);
        double r3620755 = r3620754 / r3620714;
        double r3620756 = exp(r3620755);
        double r3620757 = r3620756 - r3620732;
        double r3620758 = r3620741 ? r3620753 : r3620757;
        double r3620759 = r3620717 ? r3620739 : r3620758;
        return r3620759;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1 n) < -3.147233419817972e-08

   1. Initial program 0.8

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

   3. Applied add-cbrt-cube 0.9

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

   5. Applied add-cube-cbrt 0.9

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

   6. Applied unpow-prod-down 0.9

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

   7. Applied add-sqr-sqrt 0.9

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

   8. Applied prod-diff 0.9

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

   10. Applied add-cube-cbrt 0.9

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

   if -3.147233419817972e-08 < (/ 1 n) < 2.220885236034127e-08

   1. Initial program 45.2

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

   2. Taylor expanded around inf 32.6

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

   3. Simplified 32.5

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

   if 2.220885236034127e-08 < (/ 1 n)

   1. Initial program 25.7

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

   3. Applied add-exp-log 25.7

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

   4. Applied pow-exp 25.7

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

   5. Simplified 1.9

      \[\leadsto e^{\color{blue}{\frac{\mathsf{log1p}\left(x\right)}{n}}} - {x}^{\left(\frac{1}{n}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -3.147233419817972 \cdot 10^{-08}:\\ \;\;\;\;\sqrt[3]{\left(\left(\mathsf{fma}\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, \sqrt{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + \mathsf{fma}\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left({\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\right)\right) \cdot \left(\left(\sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}} \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right) \cdot \sqrt[3]{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\\ \mathbf{elif}\;\frac{1}{n} \le 2.220885236034127 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{1}{x \cdot n} + \frac{\log x}{x \cdot \left(n \cdot n\right)}\right) - \frac{\frac{\frac{1}{2}}{n}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{\mathsf{log1p}\left(x\right)}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))