2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.1%

Time: 3.3s

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: 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: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt x)
  (/ 0.3333333333333333 x)
  (/ (* -0.1111111111111111 (cbrt x)) (* x x))))

C

double code(double x) {
	return fma(cbrt(x), (0.3333333333333333 / x), ((-0.1111111111111111 * cbrt(x)) / (x * x)));
}

Julia

function code(x)
	return fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(-0.1111111111111111 * cbrt(x)) / Float64(x * x)))
end

Wolfram

code[x_] := N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. lift-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   10. lower-*.f64 23.8

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 23.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{\color{blue}{x} \cdot x} \]

   4. lift-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{x \cdot x} \]

   5. lift-pow.f64 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{x \cdot x} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{x \cdot x} \]

   7. lift-cbrt.f64 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{x \cdot x} \]

   8. pow2 N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{\color{blue}{2}}} \]

   9. div-add N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{{x}^{2}} + \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}} \]

   10. pow2 N/A

       \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} + \frac{\frac{-1}{9} \cdot \color{blue}{\sqrt[3]{x}}}{{x}^{2}} \]

   11. times-frac N/A

       \[\leadsto \frac{\sqrt[3]{{x}^{4}}}{x} \cdot \frac{\frac{1}{3}}{x} + \frac{\color{blue}{\frac{-1}{9} \cdot \sqrt[3]{x}}}{{x}^{2}} \]

   12. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{\frac{1}{3}}{x}}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

   13. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \frac{\color{blue}{\frac{1}{3}}}{x}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

   14. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \frac{\frac{1}{3}}{x}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

   15. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \frac{\frac{1}{3}}{x}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \frac{\frac{1}{3}}{\color{blue}{x}}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

   17. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \frac{\frac{1}{3}}{x}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}}\right) \]

6. Applied rewrites 25.0%

   \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\right) \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{\frac{1}{3}}}{x}, \frac{\frac{-1}{9} \cdot \sqrt[3]{x}}{x \cdot x}\right) \]
8. Step-by-step derivation

   1. lift-cbrt.f64 98.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\right) \]

9. Applied rewrites 98.1%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\right) \]
10. Add Preprocessing

Alternative 2: 96.4% 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. 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. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-unprod N/A

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

   7. unpow2 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. cbrt-div N/A

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

   18. metadata-eval N/A

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

   19. unpow2 N/A

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

   20. cbrt-unprod N/A

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
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 96.4

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

6. Applied rewrites 96.4%

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

Alternative 3: 92.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+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 <= 1.4e+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 <= 1.4e+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, 1.4e+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 < 1.4e154

   1. Initial program 7.0%

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

   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. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-unprod N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. cbrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. cbrt-div N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. cbrt-unprod N/A

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   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. pow2 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. pow2 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. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow-prod-down N/A

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

      8. pow2 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

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

      15. cbrt-neg-rev N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. pow2 N/A

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

      22. lift-*.f64 48.9

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

   8. Applied rewrites 48.9%

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

   if 1.4e154 < x

   1. Initial program 7.0%

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

   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. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-unprod N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. cbrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. cbrt-div N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. cbrt-unprod N/A

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   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. pow2 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. pow2 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. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 4: 92.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+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 <= 1.4e+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 <= 1.4e+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, 1.4e+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 < 1.4e154

   1. Initial program 7.0%

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

   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. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-unprod N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. cbrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. cbrt-div N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. cbrt-unprod N/A

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   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. pow2 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. pow2 N/A

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

      9. lift-*.f64 48.9

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

   6. Applied rewrites 48.9%

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

   if 1.4e154 < x

   1. Initial program 7.0%

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

   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. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. cbrt-unprod N/A

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

      7. unpow2 N/A

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

      8. associate-*r/ N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. cbrt-unprod N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. cbrt-div N/A

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

      18. metadata-eval N/A

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

      19. unpow2 N/A

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

      20. cbrt-unprod N/A

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   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. pow2 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. pow2 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. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval N/A

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

      26. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 5: 88.8% 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 7.0%

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

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. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-unprod N/A

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

   7. unpow2 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. cbrt-div N/A

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

   18. metadata-eval N/A

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

   19. unpow2 N/A

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

   20. cbrt-unprod N/A

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

4. Applied rewrites 88.8%

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

Alternative 6: 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. 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. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. cbrt-unprod N/A

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

   7. unpow2 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*r/ N/A

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

   10. unpow2 N/A

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

   11. cbrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-/l* N/A

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

   15. metadata-eval N/A

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

   16. *-rgt-identity N/A

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

   17. cbrt-div N/A

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

   18. metadata-eval N/A

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

   19. unpow2 N/A

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

   20. cbrt-unprod N/A

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
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. pow2 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. pow2 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. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. metadata-eval N/A

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

   24. lower-pow.f64 N/A

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

   25. metadata-eval N/A

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

   26. metadata-eval 88.8

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

6. Applied rewrites 88.8%

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

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

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: 98.4% 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 2025134 
(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)))