2cbrt (problem 3.3.4)

Average Error: 29.6 → 8.9

Time: 9.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.9868159492179903180542055451951455324888:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 1.423501264395558678311832436245820364987 \cdot 10^{-16}:\\ \;\;\;\;{\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

C

double f(double x) {
        double r87618 = x;
        double r87619 = 1.0;
        double r87620 = r87618 + r87619;
        double r87621 = cbrt(r87620);
        double r87622 = cbrt(r87618);
        double r87623 = r87621 - r87622;
        return r87623;
}

↓

double f(double x) {
        double r87624 = x;
        double r87625 = -0.9868159492179903;
        bool r87626 = r87624 <= r87625;
        double r87627 = 0.3333333333333333;
        double r87628 = 1.0;
        double r87629 = 2.0;
        double r87630 = pow(r87624, r87629);
        double r87631 = r87628 / r87630;
        double r87632 = cbrt(r87631);
        double r87633 = 8.0;
        double r87634 = pow(r87624, r87633);
        double r87635 = r87628 / r87634;
        double r87636 = cbrt(r87635);
        double r87637 = 0.06172839506172839;
        double r87638 = r87636 * r87637;
        double r87639 = fma(r87627, r87632, r87638);
        double r87640 = 5.0;
        double r87641 = pow(r87624, r87640);
        double r87642 = r87628 / r87641;
        double r87643 = cbrt(r87642);
        double r87644 = 0.1111111111111111;
        double r87645 = r87643 * r87644;
        double r87646 = r87639 - r87645;
        double r87647 = 1.4235012643955587e-16;
        bool r87648 = r87624 <= r87647;
        double r87649 = 1.0;
        double r87650 = r87624 + r87649;
        double r87651 = 0.3333333333333333;
        double r87652 = pow(r87650, r87651);
        double r87653 = cbrt(r87624);
        double r87654 = r87652 - r87653;
        double r87655 = r87649 + r87624;
        double r87656 = cbrt(r87655);
        double r87657 = cbrt(r87650);
        double r87658 = r87657 + r87653;
        double r87659 = r87656 * r87658;
        double r87660 = 0.6666666666666666;
        double r87661 = pow(r87624, r87660);
        double r87662 = r87659 + r87661;
        double r87663 = r87649 / r87662;
        double r87664 = r87648 ? r87654 : r87663;
        double r87665 = r87626 ? r87646 : r87664;
        return r87665;
}

Error

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if x < -0.9868159492179903

   1. Initial program 59.6

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

   2. Taylor expanded around inf 44.9

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

   3. Simplified 31.3

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

   if -0.9868159492179903 < x < 1.4235012643955587e-16

   1. Initial program 0.0

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

   3. Applied pow1/3 0.0

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

   if 1.4235012643955587e-16 < x

   1. Initial program 56.2

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

   3. Applied *-un-lft-identity 56.2

      \[\leadsto \sqrt[3]{x + 1} - \sqrt[3]{\color{blue}{1 \cdot x}}\]

   4. Applied cbrt-prod 56.2

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{x}}\]

   5. Applied add-cube-cbrt 56.4

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

   6. Applied prod-diff 56.4

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

   7. Simplified 56.4

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

   8. Simplified 56.4

      \[\leadsto \left({\left(\sqrt[3]{\sqrt[3]{x + 1}}\right)}^{3} - \sqrt[3]{x}\right) + \color{blue}{0}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 56.5

       \[\leadsto \left({\left(\sqrt[3]{\sqrt[3]{x + 1}}\right)}^{3} - \color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right) + 0\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 56.5

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left({\left(\sqrt[3]{\sqrt[3]{x + 1}}\right)}^{3} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{x + 1}}\right)}^{3} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right) \cdot \left({\left(\sqrt[3]{\sqrt[3]{x + 1}}\right)}^{3} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}} + 0\]

   13. Simplified 56.3

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}^{3}}} + 0\]
   14. Using strategy rm

   15. Applied flip3-- 56.2

       \[\leadsto \sqrt[3]{{\color{blue}{\left(\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\right)}}^{3}} + 0\]

   16. Applied cube-div 56.1

       \[\leadsto \sqrt[3]{\color{blue}{\frac{{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right)}^{3}}{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}^{3}}}} + 0\]

   17. Applied cbrt-div 56.2

       \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right)}^{3}}}{\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}^{3}}}} + 0\]

   18. Simplified 28.4

       \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt[3]{{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)\right)}^{3}}} + 0\]

   19. Simplified 4.2

       \[\leadsto \frac{1 + 0}{\color{blue}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}} + 0\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9868159492179903180542055451951455324888:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 1.423501264395558678311832436245820364987 \cdot 10^{-16}:\\ \;\;\;\;{\left(x + 1\right)}^{\frac{1}{3}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))