Average Error: 29.7 → 11.9
Time: 5.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4954988095195466 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0031086287895672454:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4954988095195466 \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 0.0031086287895672454:\\
\;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\

\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 r62173 = x;
        double r62174 = 1.0;
        double r62175 = r62173 + r62174;
        double r62176 = cbrt(r62175);
        double r62177 = cbrt(r62173);
        double r62178 = r62176 - r62177;
        return r62178;
}
↓
double f(double x) {
        double r62179 = x;
        double r62180 = -4.4954988095195466e+61;
        bool r62181 = r62179 <= r62180;
        double r62182 = 0.3333333333333333;
        double r62183 = 1.0;
        double r62184 = 2.0;
        double r62185 = pow(r62179, r62184);
        double r62186 = r62183 / r62185;
        double r62187 = 0.3333333333333333;
        double r62188 = pow(r62186, r62187);
        double r62189 = r62182 * r62188;
        double r62190 = 0.06172839506172839;
        double r62191 = 8.0;
        double r62192 = pow(r62179, r62191);
        double r62193 = r62183 / r62192;
        double r62194 = pow(r62193, r62187);
        double r62195 = r62190 * r62194;
        double r62196 = r62189 + r62195;
        double r62197 = 0.1111111111111111;
        double r62198 = 5.0;
        double r62199 = pow(r62179, r62198);
        double r62200 = r62183 / r62199;
        double r62201 = pow(r62200, r62187);
        double r62202 = r62197 * r62201;
        double r62203 = r62196 - r62202;
        double r62204 = 0.0031086287895672454;
        bool r62205 = r62179 <= r62204;
        double r62206 = 3.0;
        double r62207 = pow(r62179, r62206);
        double r62208 = 1.0;
        double r62209 = pow(r62208, r62206);
        double r62210 = r62207 + r62209;
        double r62211 = cbrt(r62210);
        double r62212 = r62179 * r62179;
        double r62213 = r62208 * r62208;
        double r62214 = r62179 * r62208;
        double r62215 = r62213 - r62214;
        double r62216 = r62212 + r62215;
        double r62217 = cbrt(r62216);
        double r62218 = r62211 / r62217;
        double r62219 = cbrt(r62179);
        double r62220 = r62218 - r62219;
        double r62221 = 0.0;
        double r62222 = r62221 + r62208;
        double r62223 = r62179 + r62208;
        double r62224 = cbrt(r62223);
        double r62225 = r62224 + r62219;
        double r62226 = r62224 * r62225;
        double r62227 = 0.6666666666666666;
        double r62228 = pow(r62179, r62227);
        double r62229 = r62226 + r62228;
        double r62230 = r62222 / r62229;
        double r62231 = r62205 ? r62220 : r62230;
        double r62232 = r62181 ? r62203 : r62231;
        return r62232;
}

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 < -4.4954988095195466e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.8

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

    if -4.4954988095195466e+61 < x < 0.0031086287895672454

    1. Initial program 4.7

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

    3. Applied flip3-+4.7

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

    4. Applied cbrt-div4.7

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

    if 0.0031086287895672454 < x

    1. Initial program 59.1

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

    3. Applied flip3--58.9

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

    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 simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4954988095195466 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.0031086287895672454:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))