2cbrt (problem 3.3.4)

Average Error: 29.3 → 0.3

Time: 8.3s

Precision: binary64

Cost: 52928

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{2}\\ t_1 := \sqrt[3]{1 + x}\\ \frac{1}{t_0 + \frac{t_1}{t_1 \cdot \left(\sqrt[3]{x} - t_1\right) - t_0} \cdot \left(-1 + x \cdot -2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 2.0)) (t_1 (cbrt (+ 1.0 x))))
   (/
    1.0
    (+ t_0 (* (/ t_1 (- (* t_1 (- (cbrt x) t_1)) t_0)) (+ -1.0 (* x -2.0)))))))

C

double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

↓

double code(double x) {
	double t_0 = pow(cbrt(x), 2.0);
	double t_1 = cbrt((1.0 + x));
	return 1.0 / (t_0 + ((t_1 / ((t_1 * (cbrt(x) - t_1)) - t_0)) * (-1.0 + (x * -2.0))));
}

Java

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.pow(Math.cbrt(x), 2.0);
	double t_1 = Math.cbrt((1.0 + x));
	return 1.0 / (t_0 + ((t_1 / ((t_1 * (Math.cbrt(x) - t_1)) - t_0)) * (-1.0 + (x * -2.0))));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(x) ^ 2.0
	t_1 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / Float64(t_0 + Float64(Float64(t_1 / Float64(Float64(t_1 * Float64(cbrt(x) - t_1)) - t_0)) * Float64(-1.0 + Float64(x * -2.0)))))
end

Wolfram

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

Derivation

1. Initial program 29.3

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

2. Applied egg-rr 28.6

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

3. Simplified 0.5

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

   [Start] 28.6	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 28.6	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 28.6	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 28.6	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 0.5	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 0.5	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 0.5	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   fma-def [=>] 0.5	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
   +-commutative [=>] 0.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

4. Applied egg-rr 0.5

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

5. Applied egg-rr 15.4

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

6. Applied egg-rr 7.5

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

7. Simplified 0.3

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

   [Start] 7.5	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{x + 1} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)}{-\left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)}} \]
   associate-/l* [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x + 1}}{\frac{-\left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)}{-\left(x + \left(x + 1\right)\right)}}}} \]
   associate-/r/ [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x + 1}}{-\left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)}} \]
   +-commutative [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{\color{blue}{1 + x}}}{-\left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   neg-sub0 [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\color{blue}{0 - \left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\color{blue}{\left(x - x\right)} - \left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\left(x - x\right) - \color{blue}{\left(\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^{2}\right)}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   associate--r+ [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\color{blue}{\left(\left(x - x\right) - \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\left(\color{blue}{0} - \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   neg-sub0 [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\color{blue}{\left(-\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   distribute-rgt-neg-in [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(-\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right)} - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-commutative [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{\color{blue}{1 + x}} \cdot \left(-\left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   sub-neg [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(\sqrt[3]{x + 1} + \left(-\sqrt[3]{x}\right)\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(\left(-\sqrt[3]{x}\right) + \sqrt[3]{x + 1}\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   neg-sub0 [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\left(\color{blue}{\left(0 - \sqrt[3]{x}\right)} + \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\left(\left(\color{blue}{\left(x - x\right)} - \sqrt[3]{x}\right) + \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   associate--r- [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(\left(x - x\right) - \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\left(\color{blue}{0} - \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   neg-sub0 [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(-\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   neg-mul-1 [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{-1 \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(0 + -1\right)} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\left(\color{blue}{\left(x - x\right)} + -1\right) \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\left(\left(x - x\right) + \color{blue}{\left(-1\right)}\right) \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   sub-neg [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(\left(x - x\right) - 1\right)} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   *-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(-\color{blue}{\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(\left(x - x\right) - 1\right)}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   distribute-rgt-neg-in [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \color{blue}{\left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(-\left(\left(x - x\right) - 1\right)\right)\right)} - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   sub-neg [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(-\color{blue}{\left(\left(x - x\right) + \left(-1\right)\right)}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-inverses [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(-\left(\color{blue}{0} + \left(-1\right)\right)\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(-\left(0 + \color{blue}{-1}\right)\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \left(-\color{blue}{-1}\right)\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right) \cdot \color{blue}{1}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   *-rgt-identity [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \color{blue}{\left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)} - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   +-commutative [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{\color{blue}{1 + x}}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\left(x + \left(x + 1\right)\right)\right)} \]
   associate-+r+ [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-\color{blue}{\left(\left(x + x\right) + 1\right)}\right)} \]
   distribute-neg-in [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \color{blue}{\left(\left(-\left(x + x\right)\right) + \left(-1\right)\right)}} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(-\left(x + x\right)\right) + \color{blue}{-1}\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(-\left(x + x\right)\right) + \color{blue}{\left(0 + -1\right)}\right)} \]
   +-inverses [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(-\left(x + x\right)\right) + \left(\color{blue}{\left(x - x\right)} + -1\right)\right)} \]
   metadata-eval [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(-\left(x + x\right)\right) + \left(\left(x - x\right) + \color{blue}{\left(-1\right)}\right)\right)} \]
   sub-neg [<=] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(-\left(x + x\right)\right) + \color{blue}{\left(\left(x - x\right) - 1\right)}\right)} \]
   +-commutative [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \color{blue}{\left(\left(\left(x - x\right) - 1\right) + \left(-\left(x + x\right)\right)\right)}} \]
   sub-neg [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\color{blue}{\left(\left(x - x\right) + \left(-1\right)\right)} + \left(-\left(x + x\right)\right)\right)} \]
   +-inverses [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(\color{blue}{0} + \left(-1\right)\right) + \left(-\left(x + x\right)\right)\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\left(0 + \color{blue}{-1}\right) + \left(-\left(x + x\right)\right)\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(\color{blue}{-1} + \left(-\left(x + x\right)\right)\right)} \]
   count-2 [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-1 + \left(-\color{blue}{2 \cdot x}\right)\right)} \]
   distribute-lft-neg-in [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-1 + \color{blue}{\left(-2\right) \cdot x}\right)} \]
   metadata-eval [=>] 0.3	\[ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2} + \frac{\sqrt[3]{1 + x}}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{1 + x}\right) - {\left(\sqrt[3]{x}\right)}^{2}} \cdot \left(-1 + \color{blue}{-2} \cdot x\right)} \]

8. Final simplification 0.3

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

Alternatives

Alternative 1
Error	24.7
Cost	39364

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	33024

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

Alternative 3
Error	8.6
Cost	32964

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{1}{t_1}} + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot t_1 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	32896

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

Alternative 5
Error	14.2
Cost	26820

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{1}{t_1}} + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \end{array} \]

Alternative 6
Error	14.2
Cost	26692

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, 1\right)}\\ \end{array} \]

Alternative 7
Error	25.3
Cost	26441

\[\begin{array}{l} \mathbf{if}\;x \leq -1.86 \cdot 10^{+15} \lor \neg \left(x \leq 2.3 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 8
Error	25.3
Cost	26185

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+15} \lor \neg \left(x \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 9
Error	25.3
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+15} \lor \neg \left(x \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 10
Error	29.3
Cost	13120

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

Alternative 11
Error	61.7
Cost	64

\[0 \]

Alternative 12
Error	31.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022354 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))