2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 97.3%

Time: 6.8s

Alternatives: 10

Speedup: 1.9×

Specification

\[x > 1 \land x < 10^{+308}\]

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: 6.7% 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: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt{\sqrt[3]{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (sqrt (pow x -1.0)) 0.3333333333333333) (sqrt (cbrt x))))

C

double code(double x) {
	return (sqrt(pow(x, -1.0)) * 0.3333333333333333) / sqrt(cbrt(x));
}

Java

public static double code(double x) {
	return (Math.sqrt(Math.pow(x, -1.0)) * 0.3333333333333333) / Math.sqrt(Math.cbrt(x));
}

Julia

function code(x)
	return Float64(Float64(sqrt((x ^ -1.0)) * 0.3333333333333333) / sqrt(cbrt(x)))
end

Wolfram

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

Derivation

1. Initial program 5.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 50.2

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

5. Applied rewrites 50.2%

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

7. Applied rewrites 92.8%

   \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot 0.3333333333333333}{\color{blue}{{x}^{0.16666666666666666}}} \]
8. Step-by-step derivation

9. Applied rewrites 98.1%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \frac{1}{3}}{\sqrt{\sqrt[3]{\color{blue}{x}}}} \]
11. Step-by-step derivation

12. Applied rewrites 98.2%

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

13. Final simplification 98.2%

    \[\leadsto \frac{\sqrt{{x}^{-1}} \cdot 0.3333333333333333}{\sqrt{\sqrt[3]{x}}} \]
14. Add Preprocessing

Alternative 2: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{x}^{-1}} \cdot 0.3333333333333333}{{x}^{0.16666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (/ (* (sqrt (pow x -1.0)) 0.3333333333333333) (pow x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (sqrt(pow(x, -1.0)) * 0.3333333333333333) / pow(x, 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (Math.sqrt(Math.pow(x, -1.0)) * 0.3333333333333333) / Math.pow(x, 0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64(Float64(sqrt((x ^ -1.0)) * 0.3333333333333333) / (x ^ 0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e155

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.7e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 92.2%

      \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot 0.3333333333333333}{\color{blue}{{x}^{0.16666666666666666}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot \frac{1}{3}}{{x}^{\frac{1}{6}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 92.2%

       \[\leadsto \frac{\sqrt{\frac{1}{x}} \cdot 0.3333333333333333}{{x}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{x}^{-1}} \cdot 0.3333333333333333}{{x}^{0.16666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1} \cdot \frac{0.3333333333333333}{{x}^{0.16666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (pow (sqrt x) -1.0) (/ 0.3333333333333333 (pow x 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = pow(sqrt(x), -1.0) * (0.3333333333333333 / pow(x, 0.16666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(Math.sqrt(x), -1.0) * (0.3333333333333333 / Math.pow(x, 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64((sqrt(x) ^ -1.0) * Float64(0.3333333333333333 / (x ^ 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.0], $MachinePrecision] * N[(0.3333333333333333 / N[Power[x, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e155

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.7e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 92.2%

      \[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\frac{0.3333333333333333}{{x}^{0.16666666666666666}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{x}\right)}^{-1} \cdot \frac{0.3333333333333333}{{x}^{0.16666666666666666}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666} \cdot 0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (/ (* (pow x -0.16666666666666666) 0.3333333333333333) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.16666666666666666) * 0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) * 0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) * 0.3333333333333333) / sqrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e155

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.7e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 6.4

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

   5. Applied rewrites 6.4%

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

   7. Applied rewrites 92.2%

      \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot 0.3333333333333333}{\color{blue}{{x}^{0.16666666666666666}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.9%

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

   11. Applied rewrites 92.2%

       \[\leadsto \frac{{x}^{-0.16666666666666666} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666} \cdot 0.3333333333333333}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) / Math.sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.7e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e155

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.7e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 6.4

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

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   9. Applied rewrites 92.2%

      \[\leadsto \frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.8e+155) {
		tmp = cbrt((pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8e+155) {
		tmp = Math.cbrt((Math.pow(x, -1.0) / x)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8e+155)
		tmp = Float64(cbrt(Float64((x ^ -1.0) / x)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.8e+155], N[(N[Power[N[(N[Power[x, -1.0], $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004e155

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 1.80000000000000004e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 6.4

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

   5. Applied rewrites 6.4%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+154)
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = cbrt(pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4e+154) {
		tmp = Math.cbrt(Math.pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+154)
		tmp = Float64(cbrt((Float64(x * x) ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+154], N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e154

   1. Initial program 7.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 96.9

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

   5. Applied rewrites 96.9%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.9%

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

   if 1.4e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 7.7

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

   5. Applied rewrites 7.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.1%

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{\sqrt[3]{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 0.3333333333333333 (sqrt x)) (sqrt (cbrt x))))

C

double code(double x) {
	return (0.3333333333333333 / sqrt(x)) / sqrt(cbrt(x));
}

Java

public static double code(double x) {
	return (0.3333333333333333 / Math.sqrt(x)) / Math.sqrt(Math.cbrt(x));
}

Julia

function code(x)
	return Float64(Float64(0.3333333333333333 / sqrt(x)) / sqrt(cbrt(x)))
end

Wolfram

code[x_] := N[(N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 50.2

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

5. Applied rewrites 50.2%

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

7. Applied rewrites 92.8%

   \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot 0.3333333333333333}{\color{blue}{{x}^{0.16666666666666666}}} \]
8. Step-by-step derivation

9. Applied rewrites 98.1%

   \[\leadsto \frac{\frac{1}{\sqrt{x}} \cdot 0.3333333333333333}{\sqrt{\sqrt[3]{x}}} \]
10. Step-by-step derivation

11. Applied rewrites 98.2%

    \[\leadsto \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{\color{blue}{\sqrt[3]{x}}}} \]
12. Add Preprocessing

Alternative 9: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x -0.6666666666666666) 0.3333333333333333))

C

double code(double x) {
	return pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.6666666666666666d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(x, -0.6666666666666666) * 0.3333333333333333

Julia

function code(x)
	return Float64((x ^ -0.6666666666666666) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.6666666666666666) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 5.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}} \]

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 50.2

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

5. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

7. Applied rewrites 89.9%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 10: 1.8% 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 5.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
4. Step-by-step derivation

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
6. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{\left(t\_0 \cdot t\_0 + \sqrt[3]{x} \cdot t\_0\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

  (- (cbrt (+ x 1.0)) (cbrt x)))