Average Error: 30.4 → 12.1
Time: 5.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.433287483387415562801378480911970005954 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.034568236329183247204492081261495428635 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.433287483387415562801378480911970005954 \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 1.034568236329183247204492081261495428635 \cdot 10^{-6}:\\
\;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\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 r69056 = x;
        double r69057 = 1.0;
        double r69058 = r69056 + r69057;
        double r69059 = cbrt(r69058);
        double r69060 = cbrt(r69056);
        double r69061 = r69059 - r69060;
        return r69061;
}
↓
double f(double x) {
        double r69062 = x;
        double r69063 = -4.4332874833874156e+61;
        bool r69064 = r69062 <= r69063;
        double r69065 = 0.3333333333333333;
        double r69066 = 1.0;
        double r69067 = 2.0;
        double r69068 = pow(r69062, r69067);
        double r69069 = r69066 / r69068;
        double r69070 = 0.3333333333333333;
        double r69071 = pow(r69069, r69070);
        double r69072 = r69065 * r69071;
        double r69073 = 0.06172839506172839;
        double r69074 = 8.0;
        double r69075 = pow(r69062, r69074);
        double r69076 = r69066 / r69075;
        double r69077 = pow(r69076, r69070);
        double r69078 = r69073 * r69077;
        double r69079 = r69072 + r69078;
        double r69080 = 0.1111111111111111;
        double r69081 = 5.0;
        double r69082 = pow(r69062, r69081);
        double r69083 = r69066 / r69082;
        double r69084 = pow(r69083, r69070);
        double r69085 = r69080 * r69084;
        double r69086 = r69079 - r69085;
        double r69087 = 1.0345682363291832e-06;
        bool r69088 = r69062 <= r69087;
        double r69089 = r69062 * r69062;
        double r69090 = 1.0;
        double r69091 = r69090 * r69090;
        double r69092 = r69089 - r69091;
        double r69093 = cbrt(r69092);
        double r69094 = r69062 - r69090;
        double r69095 = cbrt(r69094);
        double r69096 = r69093 / r69095;
        double r69097 = cbrt(r69062);
        double r69098 = r69096 - r69097;
        double r69099 = exp(r69098);
        double r69100 = log(r69099);
        double r69101 = 0.0;
        double r69102 = r69101 + r69090;
        double r69103 = r69062 + r69090;
        double r69104 = cbrt(r69103);
        double r69105 = r69104 + r69097;
        double r69106 = r69104 * r69105;
        double r69107 = 0.6666666666666666;
        double r69108 = pow(r69062, r69107);
        double r69109 = r69106 + r69108;
        double r69110 = r69102 / r69109;
        double r69111 = r69088 ? r69100 : r69110;
        double r69112 = r69064 ? r69086 : r69111;
        return r69112;
}

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 < -4.4332874833874156e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.4

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

    if -4.4332874833874156e+61 < x < 1.0345682363291832e-06

    1. Initial program 5.0

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

    3. Applied add-log-exp5.4

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

    4. Applied add-log-exp5.4

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

    5. Applied diff-log5.5

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

    6. Simplified5.0

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

    8. Applied flip-+5.0

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

    9. Applied cbrt-div5.0

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

    if 1.0345682363291832e-06 < x

    1. Initial program 58.7

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

    3. Applied flip3--58.5

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

    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. Simplified4.5

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

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.433287483387415562801378480911970005954 \cdot 10^{61}:\\ \;\;\;\;\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 1.034568236329183247204492081261495428635 \cdot 10^{-6}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))