Average Error: 29.9 → 0.6
Time: 4.2s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \left(t_0 \cdot \sqrt[3]{x} + {\left(\sqrt[3]{{t_0}^{2}}\right)}^{3}\right)} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (/
    1.0
    (+
     (pow (cbrt x) 2.0)
     (+ (* t_0 (cbrt x)) (pow (cbrt (pow t_0 2.0)) 3.0))))))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}
↓
double code(double x) {
	double t_0 = cbrt((x + 1.0));
	return 1.0 / (pow(cbrt(x), 2.0) + ((t_0 * cbrt(x)) + pow(cbrt(pow(t_0, 2.0)), 3.0)));
}

public static double code(double x) {
	return Math.cbrt((x + 1.0)) - Math.cbrt(x);
}
↓
public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	return 1.0 / (Math.pow(Math.cbrt(x), 2.0) + ((t_0 * Math.cbrt(x)) + Math.pow(Math.cbrt(Math.pow(t_0, 2.0)), 3.0)));
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end
↓
function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(1.0 / Float64((cbrt(x) ^ 2.0) + Float64(Float64(t_0 * cbrt(x)) + (cbrt((t_0 ^ 2.0)) ^ 3.0))))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
t_0 := \sqrt[3]{x + 1}\\
\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \left(t_0 \cdot \sqrt[3]{x} + {\left(\sqrt[3]{{t_0}^{2}}\right)}^{3}\right)}
\end{array}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

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

  2. Applied egg-rr29.2

    \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{\left(x + 1\right) \cdot x} + \sqrt[3]{{\left(x + 1\right)}^{2}}\right)}} \]

  3. Applied egg-rr29.3

    \[\leadsto \frac{\left(x + 1\right) - x}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{\left(x + 1\right) \cdot x} + \color{blue}{{\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{2}}\right)}^{3}}\right)} \]

  4. Taylor expanded in x around 0 15.4

    \[\leadsto \frac{\color{blue}{1}}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{\left(x + 1\right) \cdot x} + {\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{2}}\right)}^{3}\right)} \]

  5. Applied egg-rr0.6

    \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\color{blue}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x}} + {\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{2}}\right)}^{3}\right)} \]

  6. Final simplification0.6

    \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x} + {\left(\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{2}}\right)}^{3}\right)} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))