2cbrt (problem 3.3.4)

Average Error: 29.6 → 18.7

Time: 6.3s

Precision: 64

Math

\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.450208027756156638035805139753243266011 \cdot 10^{61} \lor \neg \left(x \le 3852.440338476156284741591662168502807617\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

C

double f(double x) {
        double r54730 = x;
        double r54731 = 1.0;
        double r54732 = r54730 + r54731;
        double r54733 = cbrt(r54732);
        double r54734 = cbrt(r54730);
        double r54735 = r54733 - r54734;
        return r54735;
}

↓

double f(double x) {
        double r54736 = x;
        double r54737 = -4.4502080277561566e+61;
        bool r54738 = r54736 <= r54737;
        double r54739 = 3852.4403384761563;
        bool r54740 = r54736 <= r54739;
        double r54741 = !r54740;
        bool r54742 = r54738 || r54741;
        double r54743 = 1.0;
        double r54744 = 2.0;
        double r54745 = pow(r54736, r54744);
        double r54746 = r54743 / r54745;
        double r54747 = 0.3333333333333333;
        double r54748 = pow(r54746, r54747);
        double r54749 = 0.3333333333333333;
        double r54750 = 0.06172839506172839;
        double r54751 = 8.0;
        double r54752 = pow(r54736, r54751);
        double r54753 = r54743 / r54752;
        double r54754 = pow(r54753, r54747);
        double r54755 = r54750 * r54754;
        double r54756 = 0.1111111111111111;
        double r54757 = 5.0;
        double r54758 = pow(r54736, r54757);
        double r54759 = r54743 / r54758;
        double r54760 = pow(r54759, r54747);
        double r54761 = r54756 * r54760;
        double r54762 = r54755 - r54761;
        double r54763 = fma(r54748, r54749, r54762);
        double r54764 = 1.0;
        double r54765 = r54736 + r54764;
        double r54766 = cbrt(r54765);
        double r54767 = r54766 * r54766;
        double r54768 = cbrt(r54767);
        double r54769 = cbrt(r54766);
        double r54770 = cbrt(r54736);
        double r54771 = r54770 * r54770;
        double r54772 = cbrt(r54771);
        double r54773 = cbrt(r54770);
        double r54774 = r54772 * r54773;
        double r54775 = -r54774;
        double r54776 = fma(r54768, r54769, r54775);
        double r54777 = -r54773;
        double r54778 = r54777 + r54773;
        double r54779 = r54772 * r54778;
        double r54780 = r54776 + r54779;
        double r54781 = r54773 * r54773;
        double r54782 = cbrt(r54781);
        double r54783 = cbrt(r54773);
        double r54784 = r54782 * r54783;
        double r54785 = -r54784;
        double r54786 = r54785 + r54773;
        double r54787 = r54772 * r54786;
        double r54788 = r54780 + r54787;
        double r54789 = r54742 ? r54763 : r54788;
        return r54789;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -4.4502080277561566e+61 or 3852.4403384761563 < x

   1. Initial program 60.6

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

   2. Taylor expanded around inf 36.2

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

   3. Simplified 36.2

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)}\]

   if -4.4502080277561566e+61 < x < 3852.4403384761563

   1. Initial program 4.7

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied add-log-exp 5.2

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

   4. Applied add-log-exp 5.2

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

   5. Applied diff-log 5.2

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

   6. Simplified 4.8

      \[\leadsto \log \color{blue}{\left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 4.7

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

   9. Applied cbrt-prod 4.7

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

   10. Applied *-un-lft-identity 4.7

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

   11. Applied cbrt-prod 4.7

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

   12. Applied prod-diff 4.7

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

   13. Applied exp-sum 4.7

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

   14. Applied log-prod 4.7

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

   15. Simplified 4.7

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

   16. Simplified 4.7

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

   18. Applied add-cube-cbrt 4.7

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

   19. Applied cbrt-prod 4.6

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

   21. Applied add-cube-cbrt 4.7

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

   22. Applied cbrt-prod 4.7

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

   23. Applied prod-diff 4.7

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

   24. Simplified 4.7

       \[\leadsto \left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) + \color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 18.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.450208027756156638035805139753243266011 \cdot 10^{61} \lor \neg \left(x \le 3852.440338476156284741591662168502807617\right):\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{x}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\right) + \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\left(-\sqrt[3]{\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right) + \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))