Average Error: 29.8 → 12.1
Time: 5.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.525663893489231077992208704976217314602 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 3499.391836415476063848473131656646728516:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.525663893489231077992208704976217314602 \cdot 10^{61}:\\
\;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\

\mathbf{elif}\;x \le 3499.391836415476063848473131656646728516:\\
\;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\

\end{array}
double f(double x) {
        double r133173 = x;
        double r133174 = 1.0;
        double r133175 = r133173 + r133174;
        double r133176 = cbrt(r133175);
        double r133177 = cbrt(r133173);
        double r133178 = r133176 - r133177;
        return r133178;
}
↓
double f(double x) {
        double r133179 = x;
        double r133180 = -4.525663893489231e+61;
        bool r133181 = r133179 <= r133180;
        double r133182 = 1.0;
        double r133183 = 2.0;
        double r133184 = pow(r133179, r133183);
        double r133185 = r133182 / r133184;
        double r133186 = 0.3333333333333333;
        double r133187 = pow(r133185, r133186);
        double r133188 = 0.3333333333333333;
        double r133189 = 0.06172839506172839;
        double r133190 = 8.0;
        double r133191 = pow(r133179, r133190);
        double r133192 = r133182 / r133191;
        double r133193 = pow(r133192, r133186);
        double r133194 = r133189 * r133193;
        double r133195 = 0.1111111111111111;
        double r133196 = 5.0;
        double r133197 = pow(r133179, r133196);
        double r133198 = r133182 / r133197;
        double r133199 = pow(r133198, r133186);
        double r133200 = r133195 * r133199;
        double r133201 = r133194 - r133200;
        double r133202 = fma(r133187, r133188, r133201);
        double r133203 = 3499.391836415476;
        bool r133204 = r133179 <= r133203;
        double r133205 = 1.0;
        double r133206 = r133179 + r133205;
        double r133207 = cbrt(r133206);
        double r133208 = r133207 * r133207;
        double r133209 = cbrt(r133179);
        double r133210 = r133209 * r133209;
        double r133211 = cbrt(r133210);
        double r133212 = cbrt(r133209);
        double r133213 = r133212 * r133212;
        double r133214 = r133213 * r133212;
        double r133215 = cbrt(r133214);
        double r133216 = r133211 * r133215;
        double r133217 = r133209 * r133216;
        double r133218 = r133208 - r133217;
        double r133219 = r133207 + r133209;
        double r133220 = r133218 / r133219;
        double r133221 = 7.0;
        double r133222 = pow(r133179, r133221);
        double r133223 = r133182 / r133222;
        double r133224 = pow(r133223, r133186);
        double r133225 = 0.04938271604938271;
        double r133226 = 0.6666666666666666;
        double r133227 = r133182 / r133179;
        double r133228 = pow(r133227, r133186);
        double r133229 = r133226 * r133228;
        double r133230 = 4.0;
        double r133231 = pow(r133179, r133230);
        double r133232 = r133182 / r133231;
        double r133233 = pow(r133232, r133186);
        double r133234 = r133195 * r133233;
        double r133235 = r133229 - r133234;
        double r133236 = fma(r133224, r133225, r133235);
        double r133237 = r133207 + r133214;
        double r133238 = r133236 / r133237;
        double r133239 = r133204 ? r133220 : r133238;
        double r133240 = r133181 ? r133202 : r133239;
        return r133240;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -4.525663893489231e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.2

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

    3. Simplified40.2

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

    if -4.525663893489231e+61 < x < 3499.391836415476

    1. Initial program 5.0

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

    3. Applied flip--5.0

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

    5. Applied add-cube-cbrt5.0

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

    6. Applied cbrt-prod4.9

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

    8. Applied add-cube-cbrt4.9

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

    if 3499.391836415476 < x

    1. Initial program 60.1

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

    3. Applied flip--60.1

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

    5. Applied add-cube-cbrt60.1

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

    6. Taylor expanded around inf 5.1

      \[\leadsto \frac{\color{blue}{\left(0.0493827160493827133080912972218357026577 \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]

    7. Simplified5.1

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.525663893489231077992208704976217314602 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.3333333333333333148296162562473909929395, 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 3499.391836415476063848473131656646728516:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}}, 0.0493827160493827133080912972218357026577, 0.6666666666666666296592325124947819858789 \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}} - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right)}{\sqrt[3]{x + 1} + \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))