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(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}

Error?

Try it out?

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	\[ x \cdot \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)} \]
   associate-*r/ [=>]0.0	\[ \color{blue}{\frac{x \cdot 1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)}} \]
   *-rgt-identity [=>]0.0	\[ \frac{\color{blue}{x}}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 + \sqrt[3]{1 + x}\right)} \]
   associate-+r+ [=>]0.0	\[ \frac{x}{\color{blue}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) + \sqrt[3]{1 + x}}} \]
   metadata-eval [<=]0.0	\[ \frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\left(1 - 0\right)}\right) + \sqrt[3]{1 + x}} \]
   associate--l+ [<=]0.0	\[ \frac{x}{\color{blue}{\left(\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - 0\right)} + \sqrt[3]{1 + x}} \]
   associate--r- [<=]0.0	\[ \frac{x}{\color{blue}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \left(0 - \sqrt[3]{1 + x}\right)}} \]
   neg-sub0 [<=]0.0	\[ \frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{\left(-\sqrt[3]{1 + x}\right)}} \]
   mul-1-neg [<=]0.0	\[ \frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{-1 \cdot \sqrt[3]{1 + x}}} \]
   *-commutative [<=]0.0	\[ \frac{x}{\left({\left(\sqrt[3]{1 + x}\right)}^{2} + 1\right) - \color{blue}{\sqrt[3]{1 + x} \cdot -1}} \]
   associate-+r- [<=]0.0	\[ \frac{x}{\color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(1 - \sqrt[3]{1 + x} \cdot -1\right)}} \]
   +-commutative [=>]0.0	\[ \frac{x}{\color{blue}{\left(1 - \sqrt[3]{1 + x} \cdot -1\right) + {\left(\sqrt[3]{1 + x}\right)}^{2}}} \]
   cancel-sign-sub-inv [=>]0.0	\[ \frac{x}{\color{blue}{\left(1 + \left(-\sqrt[3]{1 + x}\right) \cdot -1\right)} + {\left(\sqrt[3]{1 + x}\right)}^{2}} \]
   associate-+l+ [=>]0.0	\[ \frac{x}{\color{blue}{1 + \left(\left(-\sqrt[3]{1 + x}\right) \cdot -1 + {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}} \]

4. Final simplification0.0

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

Alternatives

Error

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))