2cbrt (problem 3.3.4)

Percentage Accurate: 53.2% → 99.6%

Time: 17.1s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Step-by-step derivation

   1. flip3-- 54.6%

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

   2. div-inv 54.6%

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

   3. rem-cube-cbrt 54.7%

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

   4. +-commutative 54.7%

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

   5. rem-cube-cbrt 56.1%

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

   6. associate--l+ 99.2%

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

3. Applied egg-rr 74.8%

   \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
4. Step-by-step derivation

   1. +-inverses 74.8%

      \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   2. metadata-eval 74.8%

      \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   3. *-lft-identity 74.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   4. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

5. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
6. Step-by-step derivation

   1. log1p-udef 74.8%

      \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   2. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   3. exp-to-pow 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   4. metadata-eval 74.8%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   5. pow-sqr 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   6. pow1/3 75.2%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. pow1/3 99.2%

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

   8. pow2 99.2%

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

7. Applied egg-rr 99.2%

   \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
8. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. flip3-+ 99.2%

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

   3. associate-*l/ 89.0%

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

   4. associate-/l* 99.2%

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

   5. +-commutative 99.2%

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

   6. rem-cube-cbrt 99.4%

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

   7. rem-cube-cbrt 99.7%

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

   8. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Step-by-step derivation

   1. flip3-- 54.6%

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

   2. div-inv 54.6%

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

   3. rem-cube-cbrt 54.7%

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

   4. +-commutative 54.7%

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

   5. rem-cube-cbrt 56.1%

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

   6. associate--l+ 99.2%

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

3. Applied egg-rr 74.8%

   \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
4. Step-by-step derivation

   1. +-inverses 74.8%

      \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   2. metadata-eval 74.8%

      \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   3. *-lft-identity 74.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   4. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

5. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
6. Step-by-step derivation

   1. log1p-udef 74.8%

      \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   2. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   3. exp-to-pow 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   4. metadata-eval 74.8%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   5. pow-sqr 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   6. pow1/3 75.2%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. pow1/3 99.2%

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

   8. pow2 99.2%

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

7. Applied egg-rr 99.2%

   \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
8. Step-by-step derivation

   1. *-commutative 99.2%

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

   2. flip3-+ 99.2%

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

   3. associate-*l/ 89.0%

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

   4. associate-/l* 99.2%

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

   5. +-commutative 99.2%

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

   6. rem-cube-cbrt 99.4%

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

   7. rem-cube-cbrt 99.7%

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

   8. +-commutative 99.7%

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

9. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\frac{x + \left(x + 1\right)}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} - \sqrt[3]{x + 1}\right)}{\sqrt[3]{x}}}}} \]
10. Step-by-step derivation

    1. associate-/r/ 99.7%

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

    2. associate-+r+ 99.7%

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

    3. +-commutative 99.7%

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

    4. +-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 60.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (/ (cbrt (fma x x -1.0)) (cbrt (+ x -1.0))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) / cbrt((x + -1.0))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) / cbrt(Float64(x + -1.0))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.5%

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

      4. +-commutative 3.5%

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

      5. rem-cube-cbrt 4.2%

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

      6. associate--l+ 98.4%

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

   3. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 46.7%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 46.7%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 46.7%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 46.7%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]

   6. Taylor expanded in x around 0 19.9%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 97.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip-+ 97.2%

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

      2. cbrt-div 97.3%

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

      3. div-inv 97.2%

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

      4. metadata-eval 97.2%

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

      5. fma-neg 97.2%

         \[\leadsto \sqrt[3]{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}} \cdot \frac{1}{\sqrt[3]{x - 1}} - \sqrt[3]{x} \]

      6. metadata-eval 97.2%

         \[\leadsto \sqrt[3]{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)} \cdot \frac{1}{\sqrt[3]{x - 1}} - \sqrt[3]{x} \]

      7. sub-neg 97.2%

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

      8. metadata-eval 97.2%

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

   3. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \frac{1}{\sqrt[3]{x + -1}}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. associate-*r/ 97.3%

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

      2. *-rgt-identity 97.3%

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

   5. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}}} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

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

Alternative 4: 60.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 1e-10)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- t_0 (pow (cbrt (cbrt x)) 3.0)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 1e-10) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = t_0 - pow(cbrt(cbrt(x)), 3.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 1e-10) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = t_0 - Math.pow(Math.cbrt(Math.cbrt(x)), 3.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 1e-10)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(t_0 - (cbrt(cbrt(x)) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 1e-10], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[Power[N[Power[N[Power[x, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 1.00000000000000004e-10

   1. Initial program 4.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 4.3%

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

      2. div-inv 4.3%

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

      3. rem-cube-cbrt 4.3%

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

      4. +-commutative 4.3%

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

      5. rem-cube-cbrt 5.0%

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

      6. associate--l+ 98.4%

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

   3. Applied egg-rr 47.1%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 47.1%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 47.1%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 47.1%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 47.1%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 47.1%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]

   6. Taylor expanded in x around 0 19.9%

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

   if 1.00000000000000004e-10 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 97.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   3. Applied egg-rr 97.8%

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

4. Final simplification 61.9%

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

Alternative 5: 86.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x 1.35e+154)
     (/ 1.0 (+ t_0 (cbrt (pow (+ 1.0 x) 2.0))))
     (/ 1.0 (+ t_0 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (t_0 + cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (t_0 + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + Math.pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(t_0 + cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64(t_0 + (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.35e+154], N[(1.0 / N[(t$95$0 + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 62.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 62.3%

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

      2. div-inv 62.3%

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

      3. rem-cube-cbrt 62.6%

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

      4. +-commutative 62.6%

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

      5. rem-cube-cbrt 64.0%

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

      6. associate--l+ 99.3%

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

   3. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 72.2%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 72.2%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 72.2%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 72.2%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   6. Step-by-step derivation

      1. log1p-udef 72.2%

         \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      2. +-commutative 72.2%

         \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      3. exp-to-pow 72.2%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      4. metadata-eval 72.2%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      5. pow-sqr 72.2%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      6. pow-prod-down 88.0%

         \[\leadsto \frac{1}{\color{blue}{{\left(\left(x + 1\right) \cdot \left(x + 1\right)\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      7. pow1/3 89.3%

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

      8. pow2 89.3%

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

   7. Applied egg-rr 89.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 2.9%

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

      4. +-commutative 2.9%

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

      5. rem-cube-cbrt 4.7%

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

      6. associate--l+ 98.4%

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

   3. Applied egg-rr 91.3%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 91.3%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 91.3%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 91.3%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 91.3%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   6. Step-by-step derivation

      1. log1p-udef 91.3%

         \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      2. +-commutative 91.3%

         \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      3. exp-to-pow 91.6%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. Applied egg-rr 91.6%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 6: 59.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (- t_0 (cbrt x))))
   (if (<= t_1 0.0) (/ 1.0 (+ (pow t_0 2.0) (cbrt x))) t_1)))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = t_0 - cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (pow(t_0, 2.0) + cbrt(x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double t_1 = t_0 - Math.cbrt(x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + Math.cbrt(x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(t_0 - cbrt(x))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + cbrt(x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 4.2%

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

      2. div-inv 4.2%

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

      3. rem-cube-cbrt 3.5%

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

      4. +-commutative 3.5%

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

      5. rem-cube-cbrt 4.2%

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

      6. associate--l+ 98.4%

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

   3. Applied egg-rr 46.7%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 46.7%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 46.7%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 46.7%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 46.7%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 46.7%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   6. Step-by-step derivation

      1. log1p-udef 46.7%

         \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      2. +-commutative 46.7%

         \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      3. exp-to-pow 46.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      4. metadata-eval 46.7%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      5. pow-sqr 46.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      6. pow1/3 47.5%

         \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      7. pow1/3 98.4%

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

      8. pow2 98.4%

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

   7. Applied egg-rr 98.4%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   8. Step-by-step derivation

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.1%

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

      3. +-commutative 98.1%

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

      4. +-commutative 98.1%

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

   9. Applied egg-rr 98.1%

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

   10. Taylor expanded in x around 0 8.9%

       \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{{\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}}} \]
   11. Step-by-step derivation

       1. metadata-eval 8.9%

          \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]

       2. pow-base-1 8.9%

          \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{1} \cdot {x}^{0.3333333333333333}} \]

       3. unpow1/3 17.7%

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

   12. Simplified 17.7%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 97.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 7: 85.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.0)
     (/ 1.0 (+ t_0 (cbrt (pow x 2.0))))
     (/ 1.0 (+ t_0 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (t_0 + cbrt(pow(x, 2.0)));
	} else {
		tmp = 1.0 / (t_0 + pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (t_0 + Math.cbrt(Math.pow(x, 2.0)));
	} else {
		tmp = 1.0 / (t_0 + Math.pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(t_0 + cbrt((x ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64(t_0 + (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(1.0 / N[(t$95$0 + N[Power[N[Power[x, 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 6.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 6.3%

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

      2. div-inv 6.3%

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

      3. rem-cube-cbrt 6.4%

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

      4. +-commutative 6.4%

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

      5. rem-cube-cbrt 7.2%

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

      6. associate--l+ 98.2%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 0.0%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 0.0%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]

   6. Taylor expanded in x around inf 56.9%

      \[\leadsto \frac{1}{\color{blue}{{\left({x}^{2}\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   7. Step-by-step derivation

      1. unpow1/3 59.5%

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

   8. Simplified 59.5%

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

   if -1 < x

   1. Initial program 69.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 69.4%

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

      2. div-inv 69.4%

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

      3. rem-cube-cbrt 69.5%

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

      4. +-commutative 69.5%

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

      5. rem-cube-cbrt 71.1%

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

      6. associate--l+ 99.5%

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

   3. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 97.6%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 97.6%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 97.6%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 97.6%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   6. Step-by-step derivation

      1. log1p-udef 97.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      3. exp-to-pow 97.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. Applied egg-rr 97.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

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

Alternative 8: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Step-by-step derivation

   1. flip3-- 54.6%

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

   2. div-inv 54.6%

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

   3. rem-cube-cbrt 54.7%

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

   4. +-commutative 54.7%

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

   5. rem-cube-cbrt 56.1%

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

   6. associate--l+ 99.2%

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

3. Applied egg-rr 74.8%

   \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
4. Step-by-step derivation

   1. +-inverses 74.8%

      \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   2. metadata-eval 74.8%

      \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   3. *-lft-identity 74.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   4. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

5. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
6. Step-by-step derivation

   1. log1p-udef 74.8%

      \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   2. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   3. exp-to-pow 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   4. metadata-eval 74.8%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{\color{blue}{\left(2 \cdot 0.3333333333333333\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   5. pow-sqr 74.8%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333} \cdot {\left(x + 1\right)}^{0.3333333333333333}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   6. pow1/3 75.2%

      \[\leadsto \frac{1}{\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. pow1/3 99.2%

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

   8. pow2 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 9: 78.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (+ t_0 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + Math.pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(t_0 + (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 6.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 6.3%

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

      2. div-inv 6.3%

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

      3. rem-cube-cbrt 6.4%

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

      4. +-commutative 6.4%

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

      5. rem-cube-cbrt 7.2%

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

      6. associate--l+ 98.2%

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

   3. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 0.0%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 0.0%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 0.0%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]

   6. Taylor expanded in x around 0 19.9%

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

   if -1 < x

   1. Initial program 69.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Step-by-step derivation

      1. flip3-- 69.4%

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

      2. div-inv 69.4%

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

      3. rem-cube-cbrt 69.5%

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

      4. +-commutative 69.5%

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

      5. rem-cube-cbrt 71.1%

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

      6. associate--l+ 99.5%

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

   3. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   4. Step-by-step derivation

      1. +-inverses 97.6%

         \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      2. metadata-eval 97.6%

         \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      3. *-lft-identity 97.6%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

      4. +-commutative 97.6%

         \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

   5. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   6. Step-by-step derivation

      1. log1p-udef 97.6%

         \[\leadsto \frac{1}{e^{\color{blue}{\log \left(1 + x\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      2. +-commutative 97.6%

         \[\leadsto \frac{1}{e^{\log \color{blue}{\left(x + 1\right)} \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

      3. exp-to-pow 97.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   7. Applied egg-rr 97.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.6666666666666666}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

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

Alternative 10: 58.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (+ 1.0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x))))))

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Step-by-step derivation

   1. flip3-- 54.6%

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

   2. div-inv 54.6%

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

   3. rem-cube-cbrt 54.7%

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

   4. +-commutative 54.7%

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

   5. rem-cube-cbrt 56.1%

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

   6. associate--l+ 99.2%

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

3. Applied egg-rr 74.8%

   \[\leadsto \color{blue}{\left(1 + \left(x - x\right)\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
4. Step-by-step derivation

   1. +-inverses 74.8%

      \[\leadsto \left(1 + \color{blue}{0}\right) \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   2. metadata-eval 74.8%

      \[\leadsto \color{blue}{1} \cdot \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   3. *-lft-identity 74.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   4. +-commutative 74.8%

      \[\leadsto \frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}\right)} \]

5. Simplified 74.8%

   \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]

6. Taylor expanded in x around 0 60.3%

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

7. Final simplification 60.3%

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

Alternative 11: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{1 + x} - \sqrt[3]{x} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.7%

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

2. Final simplification 54.7%

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

Alternative 12: 3.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in x around inf 3.6%

   \[\leadsto \color{blue}{0} \]

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 13: 49.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 54.7%

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

2. Taylor expanded in x around 0 52.0%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 52.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))