2nthrt (problem 3.4.6)

Average Error: 29.3 → 22.7

Time: 39.9s

Precision: 64

• could not determine a ground truth for program body

  1. x = 9.967111942529445e+174
  2. n = 3.0931548255617706e-64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.234230860330125218:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.097464775265143 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{\frac{-1}{3}}{{n}^{2}}, \frac{-\log x}{x}, \mathsf{fma}\left(\frac{\frac{-2}{3}}{x}, \frac{-\log x}{{n}^{2}}, \frac{1}{x \cdot n}\right)\right), -\frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

C

double f(double x, double n) {
        double r88684 = x;
        double r88685 = 1.0;
        double r88686 = r88684 + r88685;
        double r88687 = n;
        double r88688 = r88685 / r88687;
        double r88689 = pow(r88686, r88688);
        double r88690 = pow(r88684, r88688);
        double r88691 = r88689 - r88690;
        return r88691;
}

↓

double f(double x, double n) {
        double r88692 = 1.0;
        double r88693 = n;
        double r88694 = r88692 / r88693;
        double r88695 = -0.23423086033012522;
        bool r88696 = r88694 <= r88695;
        double r88697 = x;
        double r88698 = r88697 + r88692;
        double r88699 = cbrt(r88698);
        double r88700 = r88699 * r88699;
        double r88701 = pow(r88700, r88694);
        double r88702 = pow(r88699, r88694);
        double r88703 = cbrt(r88697);
        double r88704 = pow(r88703, r88694);
        double r88705 = r88703 * r88703;
        double r88706 = pow(r88705, r88694);
        double r88707 = r88704 * r88706;
        double r88708 = -r88707;
        double r88709 = fma(r88701, r88702, r88708);
        double r88710 = -r88704;
        double r88711 = r88710 + r88704;
        double r88712 = r88706 * r88711;
        double r88713 = r88709 + r88712;
        double r88714 = 3.097464775265143e-17;
        bool r88715 = r88694 <= r88714;
        double r88716 = -0.3333333333333333;
        double r88717 = 2.0;
        double r88718 = pow(r88693, r88717);
        double r88719 = r88716 / r88718;
        double r88720 = log(r88697);
        double r88721 = -r88720;
        double r88722 = r88721 / r88697;
        double r88723 = -0.6666666666666666;
        double r88724 = r88723 / r88697;
        double r88725 = r88721 / r88718;
        double r88726 = 1.0;
        double r88727 = r88697 * r88693;
        double r88728 = r88726 / r88727;
        double r88729 = fma(r88724, r88725, r88728);
        double r88730 = fma(r88719, r88722, r88729);
        double r88731 = 0.5;
        double r88732 = pow(r88697, r88717);
        double r88733 = r88732 * r88693;
        double r88734 = exp(r88733);
        double r88735 = log(r88734);
        double r88736 = r88731 / r88735;
        double r88737 = -r88736;
        double r88738 = fma(r88692, r88730, r88737);
        double r88739 = r88738 + r88712;
        double r88740 = r88717 * r88694;
        double r88741 = pow(r88698, r88740);
        double r88742 = pow(r88697, r88740);
        double r88743 = r88741 - r88742;
        double r88744 = pow(r88698, r88694);
        double r88745 = pow(r88697, r88694);
        double r88746 = r88744 + r88745;
        double r88747 = r88743 / r88746;
        double r88748 = r88715 ? r88739 : r88747;
        double r88749 = r88696 ? r88713 : r88748;
        return r88749;
}

Error

Bits error versus x

Bits error versus n

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 n) < -0.23423086033012522

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   4. Applied unpow-prod-down 0.1

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

   5. Applied add-cube-cbrt 0.1

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

   6. Applied unpow-prod-down 0.1

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

   7. Applied prod-diff 0.1

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

   8. Simplified 0.1

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

   if -0.23423086033012522 < (/ 1.0 n) < 3.097464775265143e-17

   1. Initial program 44.8

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

   3. Applied add-cube-cbrt 44.8

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

   4. Applied unpow-prod-down 44.9

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

   5. Applied add-cube-cbrt 44.9

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

   6. Applied unpow-prod-down 44.8

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

   7. Applied prod-diff 44.8

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

   8. Simplified 44.9

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

   9. Taylor expanded around inf 32.5

      \[\leadsto \color{blue}{\left(\left(1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)}{x \cdot {n}^{2}} + \left(1 \cdot \frac{1}{x \cdot n} + 1 \cdot \frac{\log \left({\left(\frac{1}{x}\right)}^{\frac{-2}{3}}\right)}{x \cdot {n}^{2}}\right)\right) - 0.5 \cdot \frac{1}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   10. Simplified 32.5

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{\frac{-1}{3}}{{n}^{2}}, \frac{-\log x}{x}, \mathsf{fma}\left(\frac{\frac{-2}{3}}{x}, \frac{-\log x}{{n}^{2}}, \frac{1}{x \cdot n}\right)\right), -\frac{0.5}{{x}^{2} \cdot n}\right)} + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]
   11. Using strategy rm

   12. Applied add-log-exp 32.5

       \[\leadsto \mathsf{fma}\left(1, \mathsf{fma}\left(\frac{\frac{-1}{3}}{{n}^{2}}, \frac{-\log x}{x}, \mathsf{fma}\left(\frac{\frac{-2}{3}}{x}, \frac{-\log x}{{n}^{2}}, \frac{1}{x \cdot n}\right)\right), -\frac{0.5}{\color{blue}{\log \left(e^{{x}^{2} \cdot n}\right)}}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\]

   if 3.097464775265143e-17 < (/ 1.0 n)

   1. Initial program 25.5

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

   3. Applied flip-- 28.5

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

   4. Simplified 28.4

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

4. Final simplification 22.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -0.234230860330125218:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{1}{n}\right)}, -{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{elif}\;\frac{1}{n} \le 3.097464775265143 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(1, \mathsf{fma}\left(\frac{\frac{-1}{3}}{{n}^{2}}, \frac{-\log x}{x}, \mathsf{fma}\left(\frac{\frac{-2}{3}}{x}, \frac{-\log x}{{n}^{2}}, \frac{1}{x \cdot n}\right)\right), -\frac{0.5}{\log \left(e^{{x}^{2} \cdot n}\right)}\right) + {\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)} \cdot \left(\left(-{\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right) + {\left(\sqrt[3]{x}\right)}^{\left(\frac{1}{n}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{\left(2 \cdot \frac{1}{n}\right)} - {x}^{\left(2 \cdot \frac{1}{n}\right)}}{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} + {x}^{\left(\frac{1}{n}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))