2cbrt (problem 3.3.4)

Percentage Accurate: 52.9% → 99.2%

Time: 16.6s

Alternatives: 14

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: 52.9% 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: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + cbrt(x))));
	} else {
		tmp = 1.0 / ((cbrt(x) * (t_0 + cbrt(x))) + cbrt(pow((1.0 + x), 2.0)));
	}
	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)) <= 0.0) {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + Math.cbrt(x))));
	} else {
		tmp = 1.0 / ((Math.cbrt(x) * (t_0 + Math.cbrt(x))) + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	}
	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((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(x)))));
	else
		tmp = Float64(1.0 / Float64(Float64(cbrt(x) * Float64(t_0 + cbrt(x))) + cbrt((Float64(1.0 + x) ^ 2.0))));
	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], 0.0], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $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 3.8%

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

      4. +-commutative 3.8%

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

      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 43.1%

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

      1. +-inverses 43.1%

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

      2. metadata-eval 43.1%

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

      3. *-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

   5. Simplified 43.1%

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

      1. log1p-udef 43.1%

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

      2. +-commutative 43.1%

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

      3. exp-to-pow 43.0%

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

      4. metadata-eval 43.0%

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

      5. pow-sqr 43.0%

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

      6. pow1/3 43.7%

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

      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. +-commutative 98.4%

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

      2. pow1/3 43.7%

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

      3. metadata-eval 43.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{0.6666666666666666}{2}\right)}} + \sqrt[3]{x}\right)} \]

      4. sqrt-pow2 43.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}} + \sqrt[3]{x}\right)} \]

      5. add-cube-cbrt 43.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}\right) \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}} + \sqrt[3]{x}\right)} \]

      6. pow3 43.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{{\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}\right)}^{3}} + \sqrt[3]{x}\right)} \]

      7. sqrt-pow2 43.7%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left({\left(\sqrt[3]{\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.6666666666666666}{2}\right)}}}\right)}^{3} + \sqrt[3]{x}\right)} \]

      8. metadata-eval 43.7%

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

      9. pow1/3 98.2%

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

      10. +-commutative 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around inf 43.7%

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

       1. unpow1/3 98.4%

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

   12. Simplified 98.4%

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

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

   1. Initial program 97.4%

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

      1. flip3-- 97.4%

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

      2. div-inv 97.4%

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

      3. rem-cube-cbrt 97.9%

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

      4. +-commutative 97.9%

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

      5. rem-cube-cbrt 99.8%

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

      6. associate--l+ 99.8%

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

   3. Applied egg-rr 96.4%

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

      1. +-inverses 96.4%

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

      2. metadata-eval 96.4%

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

      3. *-lft-identity 96.4%

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

      4. +-commutative 96.4%

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

   5. Simplified 96.4%

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

      1. log1p-udef 96.4%

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

      2. +-commutative 96.4%

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

      3. exp-to-pow 96.5%

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

      4. metadata-eval 96.5%

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

      5. pow-sqr 96.5%

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

      6. pow-prod-down 99.8%

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

      7. pow1/3 99.9%

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

      8. pow2 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

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

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 5e-6)
     (/ 1.0 (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) (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)) <= 5e-6) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + 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)) <= 5e-6)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + 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], 5e-6], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + 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)) < 5.00000000000000041e-6

   1. Initial program 5.3%

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

      1. flip3-- 5.3%

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

      2. div-inv 5.3%

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

      3. rem-cube-cbrt 5.4%

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

      4. +-commutative 5.4%

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

      5. rem-cube-cbrt 6.9%

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

      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 43.3%

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

      1. +-inverses 43.3%

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

      2. metadata-eval 43.3%

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

      3. *-lft-identity 43.3%

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

      4. +-commutative 43.3%

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

   5. Simplified 43.3%

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

      1. log1p-udef 43.3%

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

      2. +-commutative 43.3%

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

      3. exp-to-pow 43.2%

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

      4. metadata-eval 43.2%

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

      5. pow-sqr 43.2%

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

      6. pow1/3 43.9%

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

      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. +-commutative 98.4%

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

      2. pow1/3 43.9%

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

      3. metadata-eval 43.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{0.6666666666666666}{2}\right)}} + \sqrt[3]{x}\right)} \]

      4. sqrt-pow2 43.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}} + \sqrt[3]{x}\right)} \]

      5. add-cube-cbrt 43.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}\right) \cdot \sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}} + \sqrt[3]{x}\right)} \]

      6. pow3 43.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\color{blue}{{\left(\sqrt[3]{{\left(\sqrt{x + 1}\right)}^{0.6666666666666666}}\right)}^{3}} + \sqrt[3]{x}\right)} \]

      7. sqrt-pow2 43.9%

         \[\leadsto \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left({\left(\sqrt[3]{\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.6666666666666666}{2}\right)}}}\right)}^{3} + \sqrt[3]{x}\right)} \]

      8. metadata-eval 43.9%

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

      9. pow1/3 98.2%

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

      10. +-commutative 98.2%

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

   9. Applied egg-rr 98.2%

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

   10. Taylor expanded in x around inf 43.6%

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

       1. unpow1/3 97.8%

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

   12. Simplified 97.8%

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

   if 5.00000000000000041e-6 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 99.3%

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

      1. flip-+ 99.3%

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

      2. cbrt-div 99.4%

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

      3. div-inv 99.4%

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

      4. metadata-eval 99.4%

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

      5. fma-neg 99.4%

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

      6. metadata-eval 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x + -1}} - \sqrt[3]{x}\\ \end{array} \]

Alternative 3: 60.5% accurate, 0.3× speedup

Math

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

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 (cbrt (pow (+ 1.0 x) 3.0))) (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(cbrt(pow((1.0 + x), 3.0))) - cbrt(x);
	}
	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)) <= 0.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = Math.cbrt(Math.cbrt(Math.pow((1.0 + x), 3.0))) - Math.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(cbrt(cbrt((Float64(1.0 + x) ^ 3.0))) - 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[Power[N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $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 3.8%

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

      4. +-commutative 3.8%

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

      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 43.1%

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

      1. +-inverses 43.1%

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

      2. metadata-eval 43.1%

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

      3. *-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.4%

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

      1. pow1/3 95.4%

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

      2. pow-to-exp 95.2%

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

      3. +-commutative 95.2%

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

      4. log1p-def 95.2%

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

   3. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. exp-prod 95.2%

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

      2. unpow1/3 95.2%

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

   5. Simplified 95.2%

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

      1. log1p-udef 95.1%

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

      2. +-commutative 95.1%

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

      3. add-cbrt-cube 95.1%

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

      4. add-exp-log 97.4%

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

      5. pow3 97.4%

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

   7. Applied egg-rr 97.4%

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

4. Final simplification 55.7%

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

Alternative 4: 60.4% accurate, 0.3× speedup

Math

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

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 (pow t_0 3.0)) (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(pow(t_0, 3.0)) - cbrt(x);
	}
	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)) <= 0.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = Math.cbrt(Math.pow(t_0, 3.0)) - Math.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(cbrt((t_0 ^ 3.0)) - 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[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $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 3.8%

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

      4. +-commutative 3.8%

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

      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 43.1%

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

      1. +-inverses 43.1%

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

      2. metadata-eval 43.1%

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

      3. *-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.4%

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

      1. pow1/3 95.4%

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

      2. pow-to-exp 95.2%

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

      3. +-commutative 95.2%

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

      4. log1p-def 95.2%

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

   3. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. exp-prod 95.2%

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

      2. unpow1/3 95.2%

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

   5. Simplified 95.2%

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

      1. log1p-udef 95.1%

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

      2. +-commutative 95.1%

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

      3. add-exp-log 97.4%

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

      4. rem-cube-cbrt 97.5%

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

   7. Applied egg-rr 97.5%

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

4. Final simplification 55.7%

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

Alternative 5: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log t_1}\\ \end{array} \end{array} \]

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 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (exp (log 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 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = exp(log(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 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = Math.exp(Math.log(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(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = exp(log(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[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[t$95$1], $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 3.8%

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

      4. +-commutative 3.8%

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

      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 43.1%

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

      1. +-inverses 43.1%

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

      2. metadata-eval 43.1%

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

      3. *-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.4%

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

      1. add-exp-log 97.4%

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

   3. Applied egg-rr 97.4%

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

4. Final simplification 55.7%

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

Alternative 6: 87.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 t_0))
     (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 <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else 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 <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else 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 <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	elseif (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, -1.35e+154], N[(1.0 / N[(1.0 + t$95$0), $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 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.9%

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

      1. flip3-- 4.9%

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

      2. div-inv 4.9%

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

      3. rem-cube-cbrt 3.7%

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

      4. +-commutative 3.7%

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

      5. rem-cube-cbrt 4.9%

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

      6. associate--l+ 98.5%

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

   3. Applied egg-rr 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 20.0%

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

   if -1.35000000000000003e154 < x < -1

   1. Initial program 8.8%

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

      1. flip3-- 8.8%

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

      2. div-inv 8.8%

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

      3. rem-cube-cbrt 10.9%

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

      4. +-commutative 10.9%

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

      5. rem-cube-cbrt 12.1%

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

      6. associate--l+ 98.3%

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

   3. Applied egg-rr 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 90.4%

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

      1. unpow1/3 95.0%

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

   8. Simplified 95.0%

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

   if -1 < x

   1. Initial program 64.7%

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

      1. flip3-- 64.7%

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

      2. div-inv 64.7%

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

      3. rem-cube-cbrt 64.4%

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

      4. +-commutative 64.4%

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

      5. rem-cube-cbrt 65.4%

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

      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 97.4%

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

      1. +-inverses 97.4%

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

      2. metadata-eval 97.4%

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

      3. *-lft-identity 97.4%

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

      4. +-commutative 97.4%

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

   5. Simplified 97.4%

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

      1. log1p-udef 97.4%

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

      2. +-commutative 97.4%

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

      3. exp-to-pow 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 86.9%

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

Alternative 7: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (+ 1.0 (* (cbrt x) (+ t_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 / (1.0 + (cbrt(x) * (t_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 / (1.0 + (Math.cbrt(x) * (t_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(1.0 + Float64(cbrt(x) * Float64(t_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[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $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 3.8%

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

      4. +-commutative 3.8%

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

      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 43.1%

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

      1. +-inverses 43.1%

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

      2. metadata-eval 43.1%

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

      3. *-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

   5. Simplified 43.1%

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

   6. Taylor expanded in x around 0 20.0%

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

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

   1. Initial program 97.4%

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

4. Final simplification 55.7%

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

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 47.2%

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

   1. flip3-- 47.2%

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

   2. div-inv 47.2%

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

   3. rem-cube-cbrt 47.2%

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

   4. +-commutative 47.2%

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

   5. rem-cube-cbrt 48.3%

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

   6. associate--l+ 99.0%

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

3. Applied egg-rr 67.7%

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

   1. +-inverses 67.7%

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

   2. metadata-eval 67.7%

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

   3. *-lft-identity 67.7%

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

   4. +-commutative 67.7%

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

5. Simplified 67.7%

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

   1. log1p-udef 67.7%

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

   2. +-commutative 67.7%

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

   3. exp-to-pow 67.7%

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

   4. metadata-eval 67.7%

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

   5. pow-sqr 67.7%

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

   6. pow1/3 68.0%

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

   7. pow1/3 99.0%

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

   8. pow2 99.0%

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

7. Applied egg-rr 99.0%

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

8. Final simplification 99.0%

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

Alternative 9: 77.8% 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 7.2%

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

      1. flip3-- 7.2%

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

      2. div-inv 7.2%

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

      3. rem-cube-cbrt 7.9%

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

      4. +-commutative 7.9%

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

      5. rem-cube-cbrt 9.0%

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

      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 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 20.0%

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

   if -1 < x

   1. Initial program 64.7%

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

      1. flip3-- 64.7%

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

      2. div-inv 64.7%

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

      3. rem-cube-cbrt 64.4%

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

      4. +-commutative 64.4%

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

      5. rem-cube-cbrt 65.4%

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

      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 97.4%

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

      1. +-inverses 97.4%

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

      2. metadata-eval 97.4%

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

      3. *-lft-identity 97.4%

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

      4. +-commutative 97.4%

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

   5. Simplified 97.4%

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

      1. log1p-udef 97.4%

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

      2. +-commutative 97.4%

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

      3. exp-to-pow 97.3%

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 73.8%

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

Alternative 10: 52.9% 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 47.2%

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

2. Final simplification 47.2%

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

Alternative 11: 50.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- (* x 0.3333333333333333) (cbrt x))))

C

double code(double x) {
	return 1.0 + ((x * 0.3333333333333333) - cbrt(x));
}

Java

public static double code(double x) {
	return 1.0 + ((x * 0.3333333333333333) - Math.cbrt(x));
}

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * 0.3333333333333333) - cbrt(x)))
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * 0.3333333333333333), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.2%

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

   1. add-cube-cbrt 47.0%

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

   2. pow3 47.0%

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

3. Applied egg-rr 47.0%

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

4. Taylor expanded in x around 0 20.5%

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

   1. associate--l+ 20.5%

      \[\leadsto \color{blue}{1 + \left(0.3333333333333333 \cdot x - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right)} \]

   2. *-commutative 20.5%

      \[\leadsto 1 + \left(\color{blue}{x \cdot 0.3333333333333333} - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   3. metadata-eval 20.5%

      \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   4. pow-base-1 20.5%

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

   5. unpow1/3 44.6%

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

   6. *-lft-identity 44.6%

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

6. Simplified 44.6%

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

7. Final simplification 44.6%

   \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - \sqrt[3]{x}\right) \]

Alternative 12: 50.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.2%

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

   1. add-cube-cbrt 47.0%

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

   2. pow3 47.0%

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

3. Applied egg-rr 47.0%

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

4. Taylor expanded in x around 0 19.6%

   \[\leadsto \color{blue}{1 - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. metadata-eval 19.6%

      \[\leadsto 1 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333} \]

   2. pow-base-1 19.6%

      \[\leadsto 1 - \color{blue}{1} \cdot {x}^{0.3333333333333333} \]

   3. unpow1/3 44.5%

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

   4. *-lft-identity 44.5%

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

6. Simplified 44.5%

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

7. Final simplification 44.5%

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

Alternative 13: 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 47.2%

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

2. Taylor expanded in x around inf 3.7%

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

3. Final simplification 3.7%

   \[\leadsto 0 \]

Alternative 14: 49.3% 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 47.2%

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

2. Taylor expanded in x around 0 44.0%

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

3. Final simplification 44.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))