2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 97.1%

Time: 9.1s

Alternatives: 5

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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lower-/.f64 97.0

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

7. Applied rewrites 97.0%

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

8. Taylor expanded in x around 0

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

   1. lower-cbrt.f64 N/A

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

   2. lower-/.f64 97.2

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

10. Applied rewrites 97.2%

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

12. Applied rewrites 97.9%

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

Alternative 2: 97.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lower-/.f64 97.0

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

7. Applied rewrites 97.0%

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

8. Taylor expanded in x around 0

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

   1. lower-cbrt.f64 N/A

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

   2. lower-/.f64 97.2

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

10. Applied rewrites 97.2%

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

    1. lift-/.f64 N/A

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

    2. cbrt-undiv N/A

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

    3. div-inv N/A

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

    4. lift-/.f64 N/A

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

    5. cbrt-prod N/A

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

    6. lift-/.f64 N/A

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

    7. metadata-eval N/A

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

    8. associate-/r* N/A

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

    9. associate-/l/ N/A

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

    10. lift-/.f64 N/A

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

    11. cbrt-prod N/A

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

    12. lift-/.f64 N/A

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

    13. inv-pow N/A

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

    14. metadata-eval N/A

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

    15. pow-div N/A

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

    16. pow2 N/A

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

    17. lift-*.f64 N/A

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

    18. cube-unmult N/A

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

    19. lift-*.f64 N/A

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

    20. lift-*.f64 N/A

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

    21. times-frac N/A

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

    22. neg-mul-1 N/A

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

12. Applied rewrites 97.9%

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

Alternative 3: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.4

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

5. Applied rewrites 49.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 49.4

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-pow N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. pow-pow N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 89.4

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

7. Applied rewrites 89.4%

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

8. Final simplification 89.4%

   \[\leadsto 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \]
9. Add Preprocessing

Alternative 4: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 6.2%

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

   1. lift-cbrt.f64 6.2

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

   2. rem-exp-log N/A

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

   3. lift-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. log-pow N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-log.f64 6.1

      \[\leadsto \sqrt[3]{x + 1} - {\left(e^{0.3333333333333333}\right)}^{\color{blue}{\log x}} \]

4. Applied rewrites 6.1%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\log x}} \]

5. Taylor expanded in x around inf

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

   1. lower-cbrt.f64 5.2

      \[\leadsto \color{blue}{\sqrt[3]{x}} \]

7. Applied rewrites 5.2%

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

Alternative 5: 4.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.2%

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

   1. lift-cbrt.f64 6.2

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

   2. rem-exp-log N/A

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

   3. lift-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. log-pow N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-log.f64 6.1

      \[\leadsto \sqrt[3]{x + 1} - {\left(e^{0.3333333333333333}\right)}^{\color{blue}{\log x}} \]

4. Applied rewrites 6.1%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\log x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\sqrt[3]{\frac{1}{{x}^{2}}} + -1 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. mul0-lft N/A

      \[\leadsto x \cdot \color{blue}{0} \]

   4. mul0-rgt 4.2

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 4.2%

   \[\leadsto \color{blue}{0} \]
8. 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 2024219 
(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)))