Average Error: 30.4 → 11.6
Time: 6.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\
\;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\
\;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r83466 = x;
        double r83467 = 1.0;
        double r83468 = r83466 + r83467;
        double r83469 = cbrt(r83468);
        double r83470 = cbrt(r83466);
        double r83471 = r83469 - r83470;
        return r83471;
}
↓
double f(double x) {
        double r83472 = x;
        double r83473 = -4.44245620170887e+61;
        bool r83474 = r83472 <= r83473;
        double r83475 = 0.3333333333333333;
        double r83476 = 1.0;
        double r83477 = 2.0;
        double r83478 = pow(r83472, r83477);
        double r83479 = r83476 / r83478;
        double r83480 = 0.3333333333333333;
        double r83481 = pow(r83479, r83480);
        double r83482 = r83475 * r83481;
        double r83483 = 0.06172839506172839;
        double r83484 = 8.0;
        double r83485 = pow(r83472, r83484);
        double r83486 = r83476 / r83485;
        double r83487 = pow(r83486, r83480);
        double r83488 = r83483 * r83487;
        double r83489 = r83482 + r83488;
        double r83490 = 0.1111111111111111;
        double r83491 = 5.0;
        double r83492 = pow(r83472, r83491);
        double r83493 = r83476 / r83492;
        double r83494 = pow(r83493, r83480);
        double r83495 = r83490 * r83494;
        double r83496 = r83489 - r83495;
        double r83497 = 2.228167258219939e-07;
        bool r83498 = r83472 <= r83497;
        double r83499 = 1.0;
        double r83500 = r83472 + r83499;
        double r83501 = cbrt(r83500);
        double r83502 = r83501 * r83501;
        double r83503 = cbrt(r83502);
        double r83504 = cbrt(r83501);
        double r83505 = r83503 * r83504;
        double r83506 = cbrt(r83472);
        double r83507 = r83505 - r83506;
        double r83508 = log(r83507);
        double r83509 = exp(r83508);
        double r83510 = 0.0;
        double r83511 = r83510 + r83499;
        double r83512 = r83501 + r83506;
        double r83513 = r83501 * r83512;
        double r83514 = 0.6666666666666666;
        double r83515 = pow(r83472, r83514);
        double r83516 = r83513 + r83515;
        double r83517 = r83511 / r83516;
        double r83518 = r83498 ? r83509 : r83517;
        double r83519 = r83474 ? r83496 : r83518;
        return r83519;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4.44245620170887e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 38.9

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.44245620170887e+61 < x < 2.228167258219939e-07

    1. Initial program 5.0

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

    3. Applied add-cube-cbrt5.0

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

    4. Applied cbrt-prod5.0

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
    5. Using strategy rm

    6. Applied add-exp-log5.0

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

    if 2.228167258219939e-07 < x

    1. Initial program 58.3

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

    3. Applied flip3--58.2

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

    4. Simplified1.0

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

    5. Simplified4.4

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))