Average Error: 29.6 → 11.5
Time: 5.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.44550295877382874955346526182022673206 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 8.515663798864147904536456290536250435252 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.44550295877382874955346526182022673206 \cdot 10^{61}:\\
\;\;\;\;\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}}\\

\mathbf{elif}\;x \le 8.515663798864147904536456290536250435252 \cdot 10^{-6}:\\
\;\;\;\;\log \left(e^{\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \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 r60168 = x;
        double r60169 = 1.0;
        double r60170 = r60168 + r60169;
        double r60171 = cbrt(r60170);
        double r60172 = cbrt(r60168);
        double r60173 = r60171 - r60172;
        return r60173;
}
↓
double f(double x) {
        double r60174 = x;
        double r60175 = -4.445502958773829e+61;
        bool r60176 = r60174 <= r60175;
        double r60177 = 0.3333333333333333;
        double r60178 = 1.0;
        double r60179 = 2.0;
        double r60180 = pow(r60174, r60179);
        double r60181 = r60178 / r60180;
        double r60182 = 0.3333333333333333;
        double r60183 = pow(r60181, r60182);
        double r60184 = r60177 * r60183;
        double r60185 = 0.06172839506172839;
        double r60186 = 8.0;
        double r60187 = pow(r60174, r60186);
        double r60188 = r60178 / r60187;
        double r60189 = pow(r60188, r60182);
        double r60190 = r60185 * r60189;
        double r60191 = r60184 + r60190;
        double r60192 = 0.1111111111111111;
        double r60193 = 5.0;
        double r60194 = pow(r60174, r60193);
        double r60195 = r60178 / r60194;
        double r60196 = pow(r60195, r60182);
        double r60197 = r60192 * r60196;
        double r60198 = r60191 - r60197;
        double r60199 = 8.515663798864148e-06;
        bool r60200 = r60174 <= r60199;
        double r60201 = 3.0;
        double r60202 = pow(r60174, r60201);
        double r60203 = 1.0;
        double r60204 = pow(r60203, r60201);
        double r60205 = r60202 + r60204;
        double r60206 = cbrt(r60205);
        double r60207 = r60174 * r60174;
        double r60208 = r60203 * r60203;
        double r60209 = r60174 * r60203;
        double r60210 = r60208 - r60209;
        double r60211 = r60207 + r60210;
        double r60212 = cbrt(r60211);
        double r60213 = r60206 / r60212;
        double r60214 = cbrt(r60174);
        double r60215 = r60213 - r60214;
        double r60216 = exp(r60215);
        double r60217 = log(r60216);
        double r60218 = 0.0;
        double r60219 = r60218 + r60203;
        double r60220 = r60174 + r60203;
        double r60221 = cbrt(r60220);
        double r60222 = r60221 + r60214;
        double r60223 = r60221 * r60222;
        double r60224 = 0.6666666666666666;
        double r60225 = pow(r60174, r60224);
        double r60226 = r60223 + r60225;
        double r60227 = r60219 / r60226;
        double r60228 = r60200 ? r60217 : r60227;
        double r60229 = r60176 ? r60198 : r60228;
        return r60229;
}

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 < -4.445502958773829e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.1

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

    if -4.445502958773829e+61 < x < 8.515663798864148e-06

    1. Initial program 4.7

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

    3. Applied add-log-exp5.1

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

    4. Applied add-log-exp5.1

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

    5. Applied diff-log5.1

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

    6. Simplified4.7

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

    8. Applied flip3-+4.7

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

    9. Applied cbrt-div4.7

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

    if 8.515663798864148e-06 < x

    1. Initial program 58.4

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

    3. Applied flip3--58.3

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

    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 simplification11.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.44550295877382874955346526182022673206 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 8.515663798864147904536456290536250435252 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{{x}^{3} + {1}^{3}}}{\sqrt[3]{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))