# ?

Average Error: 1.8 → 0.1
Time: 10.4s
Precision: binary64
Cost: 85248

# ?

$0 \leq x \land x \leq 10^{+18}$
${\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}$
$\begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, t_0 \cdot \left(\sqrt[3]{x} - t_0\right)\right)}{\left(x + t_0 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\sqrt[3]{x} + t_0\right) - {t_0}^{4}} \end{array}$
(FPCore (x)
:precision binary64
(- (pow (+ x 1.0) (/ 1.0 3.0)) (pow x (/ 1.0 3.0))))
(FPCore (x)
:precision binary64
(let* ((t_0 (cbrt (+ x 1.0))))
(/
(fma (cbrt x) (cbrt x) (* t_0 (- (cbrt x) t_0)))
(-
(* (+ x (* t_0 (* (cbrt x) (cbrt x)))) (+ (cbrt x) t_0))
(pow t_0 4.0)))))
double code(double x) {
return pow((x + 1.0), (1.0 / 3.0)) - pow(x, (1.0 / 3.0));
}

double code(double x) {
double t_0 = cbrt((x + 1.0));
return fma(cbrt(x), cbrt(x), (t_0 * (cbrt(x) - t_0))) / (((x + (t_0 * (cbrt(x) * cbrt(x)))) * (cbrt(x) + t_0)) - pow(t_0, 4.0));
}

function code(x)
return Float64((Float64(x + 1.0) ^ Float64(1.0 / 3.0)) - (x ^ Float64(1.0 / 3.0)))
end

function code(x)
t_0 = cbrt(Float64(x + 1.0))
return Float64(fma(cbrt(x), cbrt(x), Float64(t_0 * Float64(cbrt(x) - t_0))) / Float64(Float64(Float64(x + Float64(t_0 * Float64(cbrt(x) * cbrt(x)))) * Float64(cbrt(x) + t_0)) - (t_0 ^ 4.0)))
end

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / 3.0),$MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / 3.0),$MachinePrecision]], $MachinePrecision]),$MachinePrecision]

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3],$MachinePrecision]}, N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3],$MachinePrecision] + N[(t$95$0 * N[(N[Power[x, 1/3], $MachinePrecision] - t$95$0),$MachinePrecision]), $MachinePrecision]),$MachinePrecision] / N[(N[(N[(x + N[(t$95$0 * N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3],$MachinePrecision]), $MachinePrecision]),$MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 1/3],$MachinePrecision] + t$95$0), $MachinePrecision]),$MachinePrecision] - N[Power[t$95$0, 4.0], $MachinePrecision]),$MachinePrecision]), \$MachinePrecision]]

{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}

\begin{array}{l}
t_0 := \sqrt[3]{x + 1}\\
\frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, t_0 \cdot \left(\sqrt[3]{x} - t_0\right)\right)}{\left(x + t_0 \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\sqrt[3]{x} + t_0\right) - {t_0}^{4}}
\end{array}


# Derivation?

1. Initial program 1.8

${\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}$
2. Simplified1.9

$\leadsto \color{blue}{\sqrt[3]{x + 1} - \sqrt[3]{x}}$
Proof
[Start]1.8 ${\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}$ ${\left(x + 1\right)}^{\color{blue}{0.3333333333333333}} - {x}^{\left(\frac{1}{3}\right)}$ $\color{blue}{\sqrt[3]{x + 1}} - {x}^{\left(\frac{1}{3}\right)}$ $\sqrt[3]{x + 1} - {x}^{\color{blue}{0.3333333333333333}}$ $\sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{x}}$
3. Applied egg-rr0.6

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

$\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}}$
Proof
[Start]0.6 $\left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}}$ $\frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}$ $\frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}}$ $\frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}}$ $\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}$ $\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)}$
5. Applied egg-rr0.1

$\leadsto \color{blue}{\frac{1}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}} \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}$
6. Simplified0.1

$\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right)\right)}{\left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \left(x + \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}}$
Proof
[Start]0.1 $\frac{1}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}} \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{1 + x}\right)}^{2}\right)$ $\color{blue}{\frac{1 \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}}$ $\frac{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{1 + x}\right)}^{2}}}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$ $\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)} - {\left(\sqrt[3]{1 + x}\right)}^{2}}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$ $\frac{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{x} \cdot \sqrt[3]{1 + x} - {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$ $\frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{1 + x} - {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$ $\frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{1 + x} - \color{blue}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$ $\frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \color{blue}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right)}\right)}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) \cdot \left(\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\right) - {\left(\sqrt[3]{1 + x}\right)}^{4}}$
7. Final simplification0.1

$\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right)}{\left(x + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) - {\left(\sqrt[3]{x + 1}\right)}^{4}}$

# Alternatives

Alternative 1
Error0.1
Cost39168
$\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)}$
Alternative 2
Error0.2
Cost39104
$\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}$
Alternative 3
Error0.6
Cost33088
$\frac{x + \left(1 - x\right)}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}$
Alternative 4
Error1.8
Cost13248
${\left(x + 1\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}$
Alternative 5
Error1.9
Cost13184
$\sqrt[3]{x + 1} - {x}^{0.3333333333333333}$
Alternative 6
Error1.9
Cost13120
$\sqrt[3]{x + 1} - \sqrt[3]{x}$
Alternative 7
Error6.9
Cost64
$1$

# Reproduce?

herbie shell --seed 1
(FPCore (x)
:name "pow(x+1, 1/3) - pow(x, 1/3)"
:precision binary64
:pre (and (<= 0.0 x) (<= x 1e+18))
(- (pow (+ x 1.0) (/ 1.0 3.0)) (pow x (/ 1.0 3.0))))