?

Average Error: 58.7 → 0.0
Time: 9.4s
Precision: binary64
Cost: 13632

?

$0 \leq x \land x \leq 1$
$\sqrt[3]{1 + x} - 1$
$\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{x}{1 + t_0 \cdot \left(1 + t_0\right)} \end{array}$
(FPCore (x) :precision binary64 (- (cbrt (+ 1.0 x)) 1.0))
(FPCore (x)
:precision binary64
(let* ((t_0 (cbrt (+ x 1.0)))) (/ x (+ 1.0 (* t_0 (+ 1.0 t_0))))))
double code(double x) {
return cbrt((1.0 + x)) - 1.0;
}

double code(double x) {
double t_0 = cbrt((x + 1.0));
return x / (1.0 + (t_0 * (1.0 + t_0)));
}

public static double code(double x) {
return Math.cbrt((1.0 + x)) - 1.0;
}

public static double code(double x) {
double t_0 = Math.cbrt((x + 1.0));
return x / (1.0 + (t_0 * (1.0 + t_0)));
}

function code(x)
return Float64(cbrt(Float64(1.0 + x)) - 1.0)
end

function code(x)
t_0 = cbrt(Float64(x + 1.0))
return Float64(x / Float64(1.0 + Float64(t_0 * Float64(1.0 + t_0))))
end

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

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

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


Try it out?

Results

 In Out
Enter valid numbers for all inputs

Derivation?

1. Initial program 58.7

$\sqrt[3]{1 + x} - 1$
2. Applied egg-rr0.3

$\leadsto \color{blue}{x \cdot \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)}}$
3. Simplified0.0

$\leadsto \color{blue}{\frac{x}{1 + \sqrt[3]{x + 1} \cdot \left(1 + \sqrt[3]{x + 1}\right)}}$
Proof
[Start]0.3 $x \cdot \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)}$ $\color{blue}{\frac{x \cdot 1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)}}$ $\frac{\color{blue}{x}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)}$ $\frac{x}{\color{blue}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) + \sqrt[3]{1 + x}}}$ $\frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\left(1 - 0\right)}\right) + \sqrt[3]{1 + x}}$ $\frac{x}{\color{blue}{\left(\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - 0\right)} + \sqrt[3]{1 + x}}$ $\frac{x}{\color{blue}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \left(0 - \sqrt[3]{1 + x}\right)}}$ $\frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{\left(-\sqrt[3]{1 + x}\right)}}$ $\frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{-1 \cdot \sqrt[3]{1 + x}}}$ $\frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{\sqrt[3]{1 + x} \cdot -1}}$ $\frac{x}{\color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 - \sqrt[3]{1 + x} \cdot -1\right)}}$ $\frac{x}{\color{blue}{\left(1 - \sqrt[3]{1 + x} \cdot -1\right) + {\left(\sqrt[3]{1 + x}\right)}^{2}}}$ $\frac{x}{\color{blue}{\left(1 + \left(-\sqrt[3]{1 + x}\right) \cdot -1\right)} + {\left(\sqrt[3]{1 + x}\right)}^{2}}$ $\frac{x}{\color{blue}{1 + \left(\left(-\sqrt[3]{1 + x}\right) \cdot -1 + {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}}$
4. Final simplification0.0

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

Alternatives

Alternative 1
Error0.3
Cost12992
$\mathsf{expm1}\left(0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)\right)$
Alternative 2
Error0.4
Cost7104
$\frac{x}{1 + \left(2 + \mathsf{fma}\left(x \cdot x, -0.2222222222222222, x\right)\right)}$
Alternative 3
Error1.0
Cost448
$x \cdot \left(0.3333333333333333 + x \cdot -0.1111111111111111\right)$
Alternative 4
Error0.6
Cost448
$\frac{x}{1 + \left(x + 2\right)}$
Alternative 5
Error1.6
Cost192
$x \cdot 0.3333333333333333$
Alternative 6
Error1.3
Cost192
$\frac{x}{3}$
Alternative 7
Error62.6
Cost64
$-1$
Alternative 8
Error52.7
Cost64
$x$

Reproduce?

herbie shell --seed 1
(FPCore (x)
:name "cbrt(1+x)-1"
:precision binary64
:pre (and (<= 0.0 x) (<= x 1.0))
(- (cbrt (+ 1.0 x)) 1.0))