2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 97.1%

Time: 8.6s

Alternatives: 9

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: 97.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-0.5}\right) \cdot \frac{1}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* 0.3333333333333333 (pow (cbrt x) -0.5)) (/ 1.0 (sqrt x))))

C

double code(double x) {
	return (0.3333333333333333 * pow(cbrt(x), -0.5)) * (1.0 / sqrt(x));
}

Java

public static double code(double x) {
	return (0.3333333333333333 * Math.pow(Math.cbrt(x), -0.5)) * (1.0 / Math.sqrt(x));
}

Julia

function code(x)
	return Float64(Float64(0.3333333333333333 * (cbrt(x) ^ -0.5)) * Float64(1.0 / sqrt(x)))
end

Wolfram

code[x_] := N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 43.5

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

5. Applied rewrites 43.5%

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

7. Applied rewrites 96.7%

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

9. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)} \]
10. Step-by-step derivation

11. Applied rewrites 97.4%

    \[\leadsto \frac{1}{\sqrt{x}} \cdot \left({\left(\sqrt[3]{x}\right)}^{-0.5} \cdot 0.3333333333333333\right) \]

12. Final simplification 97.4%

    \[\leadsto \left(0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-0.5}\right) \cdot \frac{1}{\sqrt{x}} \]
13. Add Preprocessing

Alternative 2: 96.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 43.5

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

5. Applied rewrites 43.5%

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

7. Applied rewrites 96.7%

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

9. Applied rewrites 97.1%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 97.3%

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

Alternative 3: 93.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right) \cdot \left(\frac{1}{x} \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (*
    (* (pow x -0.16666666666666666) 0.3333333333333333)
    (* (/ 1.0 x) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = (pow(x, -0.16666666666666666) * 0.3333333333333333) * ((1.0 / x) * sqrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) * 0.3333333333333333) * ((1.0 / x) * Math.sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) * 0.3333333333333333) * Float64(Float64(1.0 / x) * sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 92.2%

      \[\leadsto \frac{1}{\sqrt{x}} \cdot \color{blue}{\left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right) \cdot \left(\frac{1}{x} \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) / Math.sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   7. Applied rewrites 98.3%

      \[\leadsto {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 92.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (/ 1.0 (/ (pow x 0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = 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 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = 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(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(1.0 / Float64((x ^ 0.6666666666666666) / 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 0.6666666666666666], $MachinePrecision] / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   7. Applied rewrites 89.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

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

Alternative 6: 92.3% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (3.0 * 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 = 1.0 / (3.0 * 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(1.0 / Float64(3.0 * 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[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $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 8.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 88.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   7. Applied rewrites 89.1%

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

4. Final simplification 91.8%

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

Alternative 7: 92.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (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 = Math.cbrt((1.0 / (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(cbrt(Float64(1.0 / 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[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

   7. Applied rewrites 89.1%

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

Alternative 8: 88.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 43.5

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

5. Applied rewrites 43.5%

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

7. Applied rewrites 88.9%

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

Alternative 9: 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 6.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

Developer Target 1: 98.4% 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 2024234 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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