2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.4%

Time: 9.4s

Alternatives: 10

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_0, {t\_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (if (<= x 5e+14)
     (/ (- (- x -1.0) x) (fma (cbrt x) (+ (cbrt x) t_0) (pow t_0 2.0)))
     (/ (* 0.3333333333333333 (cbrt (/ -1.0 x))) (cbrt (- x))))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	double tmp;
	if (x <= 5e+14) {
		tmp = ((x - -1.0) - x) / fma(cbrt(x), (cbrt(x) + t_0), pow(t_0, 2.0));
	} else {
		tmp = (0.3333333333333333 * cbrt((-1.0 / x))) / cbrt(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / fma(cbrt(x), Float64(cbrt(x) + t_0), (t_0 ^ 2.0)));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(Float64(-1.0 / x))) / cbrt(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 5e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[(-x), 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 61.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. lower--.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

   4. Applied rewrites 99.2%

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

   if 5e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-cbrt.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 98.6%

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

Alternative 2: 96.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 96.5%

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

11. Final simplification 96.5%

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

Alternative 3: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.6%

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

Alternative 4: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.5%

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

Alternative 5: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.4%

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

Alternative 6: 92.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-cbrt.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.8%

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

   9. Applied rewrites 94.6%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 95.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg N/A

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

      5. lower-cbrt.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 8.9

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

   5. Applied rewrites 8.9%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 98.5%

      \[\leadsto \frac{0.3333333333333333}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.0%

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

Alternative 7: 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 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.4%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 88.7%

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

Alternative 8: 47.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return pow((x * x), -0.3333333333333333) * 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 * x) ** (-0.3333333333333333d0)) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-cbrt.f64 N/A

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

   6. distribute-frac-neg N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 49.5%

   \[\leadsto {\left(x \cdot x\right)}^{-0.3333333333333333} \cdot 0.3333333333333333 \]
10. Add Preprocessing

Alternative 9: 4.2% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (fma 0.3333333333333333 x (- (cbrt x))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \frac{1}{3} \cdot x\right) - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-cbrt.f64 4.3

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, 1 - \color{blue}{\sqrt[3]{x}}\right) \]

5. Applied rewrites 4.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, x, 1 - \sqrt[3]{x}\right)} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 4.3%

   \[\leadsto \mathsf{fma}\left(0.3333333333333333, x, -\sqrt[3]{x}\right) \]
9. Add Preprocessing

Alternative 10: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024357 
(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)))