2cbrt (problem 3.3.4)

Average Error: 29.3 → 0.5

Time: 9.8s

Precision: binary64

Cost: 26824

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{x} + t_0\right) \cdot t_0 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{0.3333333333333333}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= x -5e+106)
     (/ (cbrt (/ 0.037037037037037035 x)) (cbrt x))
     (if (<= x 5e+138)
       (/ 1.0 (+ (* (+ (cbrt x) t_0) t_0) (cbrt (* x x))))
       (/ 1.0 (/ (pow (cbrt x) 2.0) 0.3333333333333333))))))

C

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

↓

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if (x <= -5e+106) {
		tmp = cbrt((0.037037037037037035 / x)) / cbrt(x);
	} else if (x <= 5e+138) {
		tmp = 1.0 / (((cbrt(x) + t_0) * t_0) + cbrt((x * x)));
	} else {
		tmp = 1.0 / (pow(cbrt(x), 2.0) / 0.3333333333333333);
	}
	return tmp;
}

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.cbrt((1.0 + x));
	double tmp;
	if (x <= -5e+106) {
		tmp = Math.cbrt((0.037037037037037035 / x)) / Math.cbrt(x);
	} else if (x <= 5e+138) {
		tmp = 1.0 / (((Math.cbrt(x) + t_0) * t_0) + Math.cbrt((x * x)));
	} else {
		tmp = 1.0 / (Math.pow(Math.cbrt(x), 2.0) / 0.3333333333333333);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= -5e+106)
		tmp = Float64(cbrt(Float64(0.037037037037037035 / x)) / cbrt(x));
	elseif (x <= 5e+138)
		tmp = Float64(1.0 / Float64(Float64(Float64(cbrt(x) + t_0) * t_0) + cbrt(Float64(x * x))));
	else
		tmp = Float64(1.0 / Float64((cbrt(x) ^ 2.0) / 0.3333333333333333));
	end
	return tmp
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[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, -5e+106], N[(N[Power[N[(0.037037037037037035 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+138], N[(1.0 / N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.9999999999999998e106

   1. Initial program 61.1

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

   2. Applied egg-rr 64.0

      \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

   3. Taylor expanded in x around inf 49.2

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]

   4. Simplified 48.0

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
      Proof

      [Start] 49.2	\[ 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]
      unpow1/3 [=>] 48.0	\[ 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \]
      unpow2 [=>] 48.0	\[ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

   5. Applied egg-rr 61.1

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)} - 1} \]

   6. Simplified 48.0

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.037037037037037035}{x \cdot x}}} \]
      Proof

      [Start] 61.1	\[ e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)} - 1 \]
      expm1-def [=>] 46.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)\right)} \]
      expm1-log1p [=>] 46.8	\[ \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
      rem-cbrt-cube [<=] 47.0	\[ \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)}^{3}}} \]
      cube-prod [=>] 47.1	\[ \sqrt[3]{\color{blue}{{\left(\sqrt[3]{{x}^{-2}}\right)}^{3} \cdot {0.3333333333333333}^{3}}} \]
      rem-cube-cbrt [=>] 47.0	\[ \sqrt[3]{\color{blue}{{x}^{-2}} \cdot {0.3333333333333333}^{3}} \]
      rem-exp-log [<=] 48.0	\[ \sqrt[3]{\color{blue}{e^{\log \left({x}^{-2}\right)}} \cdot {0.3333333333333333}^{3}} \]
      log-pow [=>] 64.0	\[ \sqrt[3]{e^{\color{blue}{-2 \cdot \log x}} \cdot {0.3333333333333333}^{3}} \]
      metadata-eval [<=] 64.0	\[ \sqrt[3]{e^{\color{blue}{\left(-2\right)} \cdot \log x} \cdot {0.3333333333333333}^{3}} \]
      distribute-lft-neg-in [<=] 64.0	\[ \sqrt[3]{e^{\color{blue}{-2 \cdot \log x}} \cdot {0.3333333333333333}^{3}} \]
      exp-neg [=>] 64.0	\[ \sqrt[3]{\color{blue}{\frac{1}{e^{2 \cdot \log x}}} \cdot {0.3333333333333333}^{3}} \]
      *-commutative [=>] 64.0	\[ \sqrt[3]{\frac{1}{e^{\color{blue}{\log x \cdot 2}}} \cdot {0.3333333333333333}^{3}} \]
      exp-to-pow [=>] 48.1	\[ \sqrt[3]{\frac{1}{\color{blue}{{x}^{2}}} \cdot {0.3333333333333333}^{3}} \]
      unpow2 [=>] 48.1	\[ \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}} \cdot {0.3333333333333333}^{3}} \]
      associate-*l/ [=>] 48.1	\[ \sqrt[3]{\color{blue}{\frac{1 \cdot {0.3333333333333333}^{3}}{x \cdot x}}} \]
      metadata-eval [=>] 48.0	\[ \sqrt[3]{\frac{1 \cdot \color{blue}{0.037037037037037035}}{x \cdot x}} \]
      metadata-eval [=>] 48.0	\[ \sqrt[3]{\frac{\color{blue}{0.037037037037037035}}{x \cdot x}} \]

   7. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}} \]

   if -4.9999999999999998e106 < x < 5.00000000000000016e138

   1. Initial program 15.9

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

   2. Applied egg-rr 15.0

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

   3. Simplified 0.3

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

      [Start] 15.0	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate-*r/ [=>] 15.0	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
      *-rgt-identity [=>] 15.0	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 15.0	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate--l+ [=>] 0.3	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-inverses [=>] 0.3	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      metadata-eval [=>] 0.3	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
      fma-def [=>] 0.3	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
      +-commutative [=>] 0.3	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
      +-commutative [=>] 0.3	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

   4. Applied egg-rr 0.3

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

   5. Applied egg-rr 0.3

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

   if 5.00000000000000016e138 < x

   1. Initial program 61.0

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

   2. Applied egg-rr 61.0

      \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

   3. Taylor expanded in x around inf 55.8

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333}} \]

   4. Simplified 55.3

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
      Proof

      [Start] 55.8	\[ 0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} \]
      unpow1/3 [=>] 55.3	\[ 0.3333333333333333 \cdot \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}}} \]
      unpow2 [=>] 55.3	\[ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

   5. Applied egg-rr 61.0

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)} - 1} \]

   6. Simplified 55.3

      \[\leadsto \color{blue}{\sqrt[3]{\frac{0.037037037037037035}{x \cdot x}}} \]
      Proof

      [Start] 61.0	\[ e^{\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)} - 1 \]
      expm1-def [=>] 53.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)\right)} \]
      expm1-log1p [=>] 53.8	\[ \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
      rem-cbrt-cube [<=] 54.1	\[ \color{blue}{\sqrt[3]{{\left(\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\right)}^{3}}} \]
      cube-prod [=>] 54.1	\[ \sqrt[3]{\color{blue}{{\left(\sqrt[3]{{x}^{-2}}\right)}^{3} \cdot {0.3333333333333333}^{3}}} \]
      rem-cube-cbrt [=>] 54.0	\[ \sqrt[3]{\color{blue}{{x}^{-2}} \cdot {0.3333333333333333}^{3}} \]
      rem-exp-log [<=] 54.5	\[ \sqrt[3]{\color{blue}{e^{\log \left({x}^{-2}\right)}} \cdot {0.3333333333333333}^{3}} \]
      log-pow [=>] 54.5	\[ \sqrt[3]{e^{\color{blue}{-2 \cdot \log x}} \cdot {0.3333333333333333}^{3}} \]
      metadata-eval [<=] 54.5	\[ \sqrt[3]{e^{\color{blue}{\left(-2\right)} \cdot \log x} \cdot {0.3333333333333333}^{3}} \]
      distribute-lft-neg-in [<=] 54.5	\[ \sqrt[3]{e^{\color{blue}{-2 \cdot \log x}} \cdot {0.3333333333333333}^{3}} \]
      exp-neg [=>] 55.7	\[ \sqrt[3]{\color{blue}{\frac{1}{e^{2 \cdot \log x}}} \cdot {0.3333333333333333}^{3}} \]
      *-commutative [=>] 55.7	\[ \sqrt[3]{\frac{1}{e^{\color{blue}{\log x \cdot 2}}} \cdot {0.3333333333333333}^{3}} \]
      exp-to-pow [=>] 55.3	\[ \sqrt[3]{\frac{1}{\color{blue}{{x}^{2}}} \cdot {0.3333333333333333}^{3}} \]
      unpow2 [=>] 55.3	\[ \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}} \cdot {0.3333333333333333}^{3}} \]
      associate-*l/ [=>] 55.3	\[ \sqrt[3]{\color{blue}{\frac{1 \cdot {0.3333333333333333}^{3}}{x \cdot x}}} \]
      metadata-eval [=>] 55.3	\[ \sqrt[3]{\frac{1 \cdot \color{blue}{0.037037037037037035}}{x \cdot x}} \]
      metadata-eval [=>] 55.3	\[ \sqrt[3]{\frac{\color{blue}{0.037037037037037035}}{x \cdot x}} \]

   7. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\frac{1}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{0.3333333333333333}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{\left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x} + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{0.3333333333333333}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	39684

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 10^{-5}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right) + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	39168

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

Alternative 3
Error	0.6
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 4
Error	0.5
Cost	32896

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

Alternative 5
Error	0.8
Cost	26308

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	0.8
Cost	20360

\[\begin{array}{l} \mathbf{if}\;x \leq -35000000:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 42000000:\\ \;\;\;\;\sqrt[3]{\frac{1 + {x}^{3}}{1 + \left(x \cdot x - x\right)}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\sqrt{0.037037037037037035}}{x}}\right)}^{2}\\ \end{array} \]

Alternative 7
Error	0.8
Cost	19720

\[\begin{array}{l} \mathbf{if}\;x \leq -35000000:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 42000000:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt[3]{1 + x} - \sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\sqrt{0.037037037037037035}}{x}}\right)}^{2}\\ \end{array} \]

Alternative 8
Error	0.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -35000000:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 42000000:\\ \;\;\;\;\frac{1}{\frac{1}{\sqrt[3]{1 + x} - \sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{0.3333333333333333}}\\ \end{array} \]

Alternative 9
Error	0.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;x \leq -35000000:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 42000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{0.3333333333333333}}\\ \end{array} \]

Alternative 10
Error	0.8
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq -35000000:\\ \;\;\;\;\frac{\sqrt[3]{\frac{0.037037037037037035}{x}}}{\sqrt[3]{x}}\\ \mathbf{elif}\;x \leq 42000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\\ \end{array} \]

Alternative 11
Error	3.1
Cost	13321

\[\begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 0.24\right):\\ \;\;\;\;\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.6666666666666666}\\ \end{array} \]

Alternative 12
Error	17.7
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 0.24\right):\\ \;\;\;\;\sqrt[3]{\frac{0.037037037037037035}{x} \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.6666666666666666}\\ \end{array} \]

Alternative 13
Error	18.0
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -0.084 \lor \neg \left(x \leq 0.24\right):\\ \;\;\;\;\sqrt[3]{\frac{0.037037037037037035}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.6666666666666666}\\ \end{array} \]

Alternative 14
Error	61.7
Cost	64

\[0 \]

Alternative 15
Error	31.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023045 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))