2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 96.5%

Time: 2.9s

Alternatives: 7

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 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 6.9%

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

2. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-neg-rev N/A

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

   7. metadata-eval N/A

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

   8. pow-flip N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

4. Applied rewrites 88.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-cbrt.f64 96.5

      \[\leadsto \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \]

6. Applied rewrites 96.5%

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

Alternative 2: 92.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{-x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{x}^{-0.6666666666666666}}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ -1.0 (cbrt (- (* x x)))) 0.3333333333333333)
   (* (/ 1.0 (/ 1.0 (pow x -0.6666666666666666))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / (1.0 / pow(x, -0.6666666666666666))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / Math.cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / (1.0 / Math.pow(x, -0.6666666666666666))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(-1.0 / cbrt(Float64(-Float64(x * x)))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 / (x ^ -0.6666666666666666))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(-1.0 / N[Power[(-N[(x * x), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[(1.0 / N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. pow1/3 N/A

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

      16. unpow2 N/A

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

      17. pow-prod-down N/A

          \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. pow-flip N/A

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

      21. lower-pow.f64 N/A

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

      22. metadata-eval 88.9

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

   6. Applied rewrites 88.9%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\frac{-2}{3}} \cdot \frac{1}{3} \]

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

         \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. rem-cbrt-cube N/A

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

      16. lower-/.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. pow1/3 N/A

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

      16. unpow2 N/A

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

      17. pow-prod-down N/A

          \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. pow-flip N/A

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

      21. lower-pow.f64 N/A

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

      22. metadata-eval 88.9

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

   6. Applied rewrites 88.9%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\frac{-2}{3}} \cdot \frac{1}{3} \]

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

         \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. mult-flip N/A

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

      12. cbrt-prod N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. rem-cbrt-cube N/A

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

      17. metadata-eval N/A

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

      18. pow-flip N/A

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

      19. *-commutative N/A

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

      20. mult-flip-rev N/A

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

      21. unpow2 N/A

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

      22. cbrt-unprod N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. division-flip N/A

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

      26. lower-special-/ N/A

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

      27. lower-/.f64 N/A

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

      28. lower-special-/ N/A

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

      29. lower-/.f64 N/A

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

   8. Applied rewrites 88.9%

      \[\leadsto \frac{1}{\frac{1}{{x}^{-0.6666666666666666}}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sqrt[3]{-x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ -1.0 (cbrt (- (* x x)))) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (-1.0 / Math.cbrt(-(x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(-1.0 / cbrt(Float64(-Float64(x * x)))) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(-1.0 / N[Power[(-N[(x * x), $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. pow1/3 N/A

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

      16. unpow2 N/A

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

      17. pow-prod-down N/A

          \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. pow-flip N/A

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

      21. lower-pow.f64 N/A

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

      22. metadata-eval 88.9

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

   6. Applied rewrites 88.9%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto {x}^{\frac{-2}{3}} \cdot \frac{1}{3} \]

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow-prod-up N/A

         \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      6. pow-prod-down N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. frac-2neg N/A

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

      12. metadata-eval N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. rem-cbrt-cube N/A

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

      16. lower-/.f64 N/A

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

      17. lower-cbrt.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. pow1/3 N/A

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

      16. unpow2 N/A

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

      17. pow-prod-down N/A

          \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. pow-flip N/A

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

      21. lower-pow.f64 N/A

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

      22. metadata-eval 88.9

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

   6. Applied rewrites 88.9%

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

Alternative 4: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. unpow2 N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 51.2

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

   6. Applied rewrites 51.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 6.9%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-neg-rev N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. unpow2 N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. pow1/3 N/A

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

      16. unpow2 N/A

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

      17. pow-prod-down N/A

          \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

      18. pow-prod-up N/A

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

      19. metadata-eval N/A

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

      20. pow-flip N/A

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

      21. lower-pow.f64 N/A

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

      22. metadata-eval 88.9

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

   6. Applied rewrites 88.9%

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

Alternative 5: 88.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

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

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 = 0.3333333333333333d0 / (x ** 0.6666666666666666d0)
end function

Java

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

Python

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

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-neg-rev N/A

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

   7. metadata-eval N/A

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

   8. pow-flip N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

4. Applied rewrites 88.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Add Preprocessing

Alternative 6: 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

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 6.9%

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

2. Taylor expanded in x around inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-neg-rev N/A

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

   7. metadata-eval N/A

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

   8. pow-flip N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

4. Applied rewrites 88.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. metadata-eval N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow-prod-down N/A

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

   8. unpow2 N/A

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

   9. pow1/3 N/A

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

   10. cbrt-div N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. cbrt-div N/A

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

   14. metadata-eval N/A

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

   15. pow1/3 N/A

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

   16. unpow2 N/A

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

   17. pow-prod-down N/A

       \[\leadsto \frac{1}{{x}^{\frac{1}{3}} \cdot {x}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

   18. pow-prod-up N/A

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

   19. metadata-eval N/A

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

   20. pow-flip N/A

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

   21. lower-pow.f64 N/A

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

   22. metadata-eval 88.9

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

6. Applied rewrites 88.9%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
7. Add Preprocessing

Alternative 7: 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 6.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
5. 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 2025132 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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