2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 97.0%

Time: 8.4s

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: 7.1% 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: 97.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-lowering-*.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. cbrt-lowering-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. /-lowering-/.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. *-lowering-*.f64 49.4

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

5. Simplified 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. clear-num N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. cbrt-lowering-cbrt.f64 N/A

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

   8. pow1/3 N/A

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

   9. cbrt-div N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f64 N/A

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

   12. cbrt-lowering-cbrt.f64 97.6

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

7. Applied egg-rr 97.6%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. sqr-neg N/A

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

   4. +-lft-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\left(0 - \sqrt[3]{x}\right)}} \]

   6. flip-- N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{\color{blue}{0} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}} \]

   8. /-rgt-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{0 - \color{blue}{\frac{\sqrt[3]{x}}{1}} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}} \]

   9. associate-/r/ N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{0 - \color{blue}{\frac{\sqrt[3]{x}}{\frac{1}{\sqrt[3]{x}}}}}{0 + \sqrt[3]{x}}} \]

   10. neg-sub0 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{\sqrt[3]{x}}{\frac{1}{\sqrt[3]{x}}}\right)}}{0 + \sqrt[3]{x}}} \]

   11. +-lft-identity N/A

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

   12. associate-*r/ N/A

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

9. Applied egg-rr 98.2%

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

    1. un-div-inv N/A

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

    2. associate-/r/ N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. *-lowering-*.f64 N/A

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

    6. metadata-eval N/A

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

    7. frac-2neg N/A

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

    8. /-lowering-/.f64 N/A

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

    9. cbrt-lowering-cbrt.f64 98.3

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

11. Applied egg-rr 98.3%

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

Alternative 2: 97.0% 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 5.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-lowering-*.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. cbrt-lowering-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. /-lowering-/.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. *-lowering-*.f64 49.4

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

5. Simplified 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. clear-num N/A

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

   5. /-lowering-/.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. cbrt-lowering-cbrt.f64 N/A

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

   8. pow1/3 N/A

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

   9. cbrt-div N/A

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

   10. metadata-eval N/A

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

   11. /-lowering-/.f64 N/A

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

   12. cbrt-lowering-cbrt.f64 97.6

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

7. Applied egg-rr 97.6%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. sqr-neg N/A

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

   4. +-lft-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\left(0 + \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)}} \]

   5. sub-neg N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\left(0 - \sqrt[3]{x}\right)}} \]

   6. flip-- N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{\color{blue}{0} - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}} \]

   8. /-rgt-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{0 - \color{blue}{\frac{\sqrt[3]{x}}{1}} \cdot \sqrt[3]{x}}{0 + \sqrt[3]{x}}} \]

   9. associate-/r/ N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{0 - \color{blue}{\frac{\sqrt[3]{x}}{\frac{1}{\sqrt[3]{x}}}}}{0 + \sqrt[3]{x}}} \]

   10. neg-sub0 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{\sqrt[3]{x}}{\frac{1}{\sqrt[3]{x}}}\right)}}{0 + \sqrt[3]{x}}} \]

   11. +-lft-identity N/A

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

   12. associate-*r/ N/A

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

9. Applied egg-rr 98.2%

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

    1. clear-num N/A

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

    2. /-lowering-/.f64 N/A

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

    3. cbrt-lowering-cbrt.f64 98.3

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

11. Applied egg-rr 98.3%

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

Alternative 3: 88.7% 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

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 5.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-lowering-*.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. cbrt-lowering-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. /-lowering-/.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. *-lowering-*.f64 49.4

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

5. Simplified 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. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. un-div-inv N/A

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

   7. /-lowering-/.f64 N/A

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

   8. pow1/3 N/A

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

   9. pow1/3 N/A

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

   10. pow-sqr N/A

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

   11. pow-lowering-pow.f64 N/A

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

   12. metadata-eval 89.8

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

7. Applied egg-rr 89.8%

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

Alternative 4: 88.7% 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 5.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-lowering-*.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. cbrt-lowering-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. /-lowering-/.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. *-lowering-*.f64 49.4

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

5. Simplified 49.4%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval 89.8

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

7. Applied egg-rr 89.8%

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

8. Final simplification 89.8%

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

Alternative 5: 5.3% 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 5.6%

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

3. Taylor expanded in x around 0

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

   1. --lowering--.f64 N/A

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

   2. cbrt-lowering-cbrt.f64 1.8

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

5. Simplified 1.8%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. cbrt-lowering-cbrt.f64 1.8

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

8. Simplified 1.8%

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

   1. pow1/3 N/A

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

   2. pow-lowering-pow.f64 1.8

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

10. Applied egg-rr 1.8%

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

12. Applied egg-rr 5.2%

    \[\leadsto \color{blue}{\sqrt[3]{x}} \]
13. 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 2024198 
(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)))