Average Error: 30.2 → 0.5
Time: 9.8s
Precision: binary64
Cost: 32896
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ t_0 (cbrt x)))))))
double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}
↓
double code(double x) {
	double t_0 = cbrt((1.0 + x));
	return 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (t_0 + cbrt(x))));
}

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((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
}

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end
↓
function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))))
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[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.2

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

  2. Applied egg-rr29.6

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

  3. Taylor expanded in x around 0 0.5

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

  4. Final simplification0.5

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

Alternatives

Alternative 1
Error13.0
Cost32968
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.2929836526227745 \cdot 10^{+158}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(t_1\right)}\\ \mathbf{elif}\;x \leq 1.7875739492581274 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{\log t_1}}\\ \end{array} \]
Alternative 2
Error13.0
Cost32968
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1.2929836526227745 \cdot 10^{+158}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot t_1\right)}\\ \mathbf{elif}\;x \leq 1.7875739492581274 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + t_1 \cdot e^{\log \left(\sqrt[3]{x}\right)}}\\ \end{array} \]
Alternative 3
Error13.0
Cost32644
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.2929836526227745 \cdot 10^{+158}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(t_1\right)}\\ \mathbf{elif}\;x \leq 1.7875739492581274 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + t_1}\\ \end{array} \]
Alternative 4
Error13.0
Cost26888
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{if}\;x \leq -1.2929836526227745 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7875739492581274 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error26.5
Cost19904
\[\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
Alternative 6
Error30.2
Cost13120
\[\sqrt[3]{1 + x} - \sqrt[3]{x} \]
Alternative 7
Error31.8
Cost6592
\[1 - \sqrt[3]{x} \]
Alternative 8
Error32.2
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022295 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))