Average Error: 29.9 → 12.0
Time: 4.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.45593027296653583 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 2.2954140183814659 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{{x}^{3} - {1}^{3}}}, \sqrt[3]{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}, -\sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.45593027296653583 \cdot 10^{61}:\\
\;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\

\mathbf{elif}\;x \le 2.2954140183814659 \cdot 10^{-10}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\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}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {x}^{\frac{2}{3}}\right)}\\

\end{array}
double code(double x) {
	return (cbrt((x + 1.0)) - cbrt(x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -4.455930272966536e+61)) {
		VAR = fma(pow((1.0 / pow(x, 2.0)), 0.3333333333333333), 0.3333333333333333, ((0.06172839506172839 * pow((1.0 / pow(x, 8.0)), 0.3333333333333333)) - (0.1111111111111111 * pow((1.0 / pow(x, 5.0)), 0.3333333333333333))));
	} else {
		double VAR_1;
		if ((x <= 2.295414018381466e-10)) {
			VAR_1 = fma((cbrt(((x * x) - (1.0 * 1.0))) / cbrt((pow(x, 3.0) - pow(1.0, 3.0)))), cbrt(((x * x) + ((1.0 * 1.0) + (x * 1.0)))), -cbrt(x));
		} else {
			VAR_1 = ((0.0 + 1.0) / fma(cbrt((x + 1.0)), (cbrt((x + 1.0)) + cbrt(x)), pow(x, 0.6666666666666666)));
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 < -4.455930272966536e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 41.3

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

    3. Simplified41.3

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

    if -4.455930272966536e+61 < x < 2.295414018381466e-10

    1. Initial program 4.9

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

    3. Applied flip-+4.9

      \[\leadsto \sqrt[3]{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \sqrt[3]{x}\]

    4. Applied cbrt-div4.8

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

    6. Applied flip3--4.8

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

    7. Applied cbrt-div4.8

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

    8. Applied associate-/r/4.8

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

    9. Applied fma-neg4.8

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

    if 2.295414018381466e-10 < x

    1. Initial program 57.6

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

    3. Applied flip3--57.5

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

    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. Simplified4.3

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.45593027296653583 \cdot 10^{61}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}}, 0.333333333333333315, 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\right)\\ \mathbf{elif}\;x \le 2.2954140183814659 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{{x}^{3} - {1}^{3}}}, \sqrt[3]{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}, -\sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))