2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.4%

Time: 5.9s

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: 7.0% 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: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0 + \sqrt[3]{x}, {t\_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= x 4e+15)
     (/ (- (+ 1.0 x) x) (fma (cbrt x) (+ t_0 (cbrt x)) (pow t_0 2.0)))
     (* (/ (cbrt (pow x -1.0)) (cbrt x)) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if (x <= 4e+15) {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (t_0 + cbrt(x)), pow(t_0, 2.0));
	} else {
		tmp = (cbrt(pow(x, -1.0)) / cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 4e+15)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(t_0 + cbrt(x)), (t_0 ^ 2.0)));
	else
		tmp = Float64(Float64(cbrt((x ^ -1.0)) / cbrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 4e+15], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4e15

   1. Initial program 55.4%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 53.6

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

   4. Applied rewrites 53.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow1/3 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. rem-cube-cbrt N/A

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

      9. lift-cbrt.f64 N/A

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

      10. rem-cube-cbrt N/A

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

      11. lower--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-rgt-out N/A

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

      17. +-commutative N/A

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

   6. Applied rewrites 98.9%

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

   if 4e15 < x

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 52.7

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

   5. Applied rewrites 52.7%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{-1}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 96.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 96.4%

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

13. Final simplification 96.4%

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

Alternative 3: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4e+155)
   (* (cbrt (/ (pow x -1.0) x)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4e+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 <= 2.4e+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 <= 2.4e+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, 2.4e+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 < 2.40000000000000021e155

   1. Initial program 8.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

   if 2.40000000000000021e155 < 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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 6.0

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

   5. Applied rewrites 6.0%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   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 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{\frac{{x}^{-1}}{x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

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. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

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

7. Applied rewrites 96.4%

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

Alternative 5: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+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 <= 1.32e+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 <= 1.32e+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, 1.32e+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 < 1.31999999999999998e154

   1. Initial program 9.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.9%

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

   if 1.31999999999999998e154 < 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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 8.2

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

   5. Applied rewrites 8.2%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   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 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4e+155)
   (* (cbrt (pow x -2.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4e+155) {
		tmp = cbrt(pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4e+155) {
		tmp = Math.cbrt(Math.pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4e+155)
		tmp = Float64(cbrt((x ^ -2.0)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4e+155], N[(N[Power[N[Power[x, -2.0], $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 < 2.40000000000000021e155

   1. Initial program 8.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 95.0%

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   if 2.40000000000000021e155 < 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. unpow2 N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 6.0

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

   5. Applied rewrites 6.0%

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

Alternative 7: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.3333333333333333 (pow (cbrt x) 2.0)))

C

double code(double x) {
	return 0.3333333333333333 / pow(cbrt(x), 2.0);
}

Java

public static double code(double x) {
	return 0.3333333333333333 / Math.pow(Math.cbrt(x), 2.0);
}

Julia

function code(x)
	return Float64(0.3333333333333333 / (cbrt(x) ^ 2.0))
end

Wolfram

code[x_] := N[(0.3333333333333333 / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

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

7. Applied rewrites 96.4%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
8. Add Preprocessing

Alternative 8: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow (cbrt x) -2.0) 0.3333333333333333))

C

double code(double x) {
	return pow(cbrt(x), -2.0) * 0.3333333333333333;
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
}

Julia

function code(x)
	return Float64((cbrt(x) ^ -2.0) * 0.3333333333333333)
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

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. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

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

7. Applied rewrites 96.4%

   \[\leadsto {\left(\sqrt[3]{x}\right)}^{-2} \cdot \color{blue}{0.3333333333333333} \]
8. Add Preprocessing

Alternative 9: 88.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 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. unpow2 N/A

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

   16. unpow2 N/A

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

   17. associate-/r* N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 53.2

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

5. Applied rewrites 53.2%

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

7. Applied rewrites 88.8%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
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 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024364 
(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)))