2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 97.4%

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: 6.7% 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: 97.4% accurate, 1.7× speedup

Math

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

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)) / Float64(x / cbrt(x)))
end

Wolfram

code[x_] := N[((--0.3333333333333333) / N[(x / N[Power[x, 1/3], $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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 49.9

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

5. Applied rewrites 49.9%

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

7. Applied rewrites 89.2%

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

9. Applied rewrites 97.7%

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

10. Final simplification 97.7%

    \[\leadsto \frac{--0.3333333333333333}{\frac{x}{\sqrt[3]{x}}} \]
11. Add Preprocessing

Alternative 2: 97.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[Power[(-x), 1/3], $MachinePrecision] * N[(0.3333333333333333 / (-x)), $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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 49.9

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

5. Applied rewrites 49.9%

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

7. Applied rewrites 97.0%

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

9. Applied rewrites 97.0%

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

11. Applied rewrites 97.6%

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

12. Final simplification 97.6%

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

Alternative 3: 97.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Power[x, 1/3], $MachinePrecision] / x), $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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 49.9

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

5. Applied rewrites 49.9%

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

7. Applied rewrites 89.2%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
8. Step-by-step derivation

9. Applied rewrites 97.6%

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

Alternative 4: 89.0% 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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 49.9

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

5. Applied rewrites 49.9%

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

7. Applied rewrites 89.2%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
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 6.3%

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

   1. rem-cube-cbrt N/A

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

   2. pow1/3 N/A

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

   3. pow-to-exp N/A

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

   4. pow-exp N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. rem-cube-cbrt N/A

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

   10. lift-cbrt.f64 N/A

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

   11. pow-to-exp N/A

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

   12. rem-log-exp N/A

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

   13. lower-*.f64 N/A

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

   14. rem-log-exp N/A

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

   15. pow-to-exp N/A

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

   16. lift-cbrt.f64 N/A

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

   17. rem-cube-cbrt N/A

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-log1p.f64 4.5

       \[\leadsto \sqrt[3]{{\left(e^{3}\right)}^{\left(\color{blue}{\mathsf{log1p}\left(x\right)} \cdot 0.3333333333333333\right)}} - \sqrt[3]{x} \]

4. Applied rewrites 4.5%

   \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{3}\right)}^{\left(\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333\right)}}} - \sqrt[3]{x} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{0} \]
6. Step-by-step derivation

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 2024331 
(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)))