2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 99.0%

Time: 8.2s

Alternatives: 4

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 10^{-10}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \sqrt[3]{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0 + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 1e-10)
     (/ (* -0.3333333333333333 (cbrt x)) (- x))
     (/
      (- (+ 1.0 x) x)
      (fma
       (cbrt x)
       (+ t_0 (cbrt x))
       (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 1e-10) {
		tmp = (-0.3333333333333333 * cbrt(x)) / -x;
	} else {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (t_0 + cbrt(x)), exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 1e-10)
		tmp = Float64(Float64(-0.3333333333333333 * cbrt(x)) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(t_0 + cbrt(x)), exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 1e-10], N[(N[(-0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 1.00000000000000004e-10

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]

   5. Applied rewrites 20.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 68.9%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{-x} \cdot \color{blue}{\frac{-1}{x}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   12. Applied rewrites 99.1%

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

   if 1.00000000000000004e-10 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 63.9%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 63.5

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

   4. Applied rewrites 63.5%

      \[\leadsto \sqrt[3]{\color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, 1\right)}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. rem-square-sqrt N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cbrt.f64 N/A

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

      12. clear-num N/A

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

      13. flip-+ N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 63.2

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

   6. Applied rewrites 63.2%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-/.f64 N/A

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

   8. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 10^{-10}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \sqrt[3]{x}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 1.7× speedup

Math

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

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(Float64(-0.3333333333333333 * cbrt(x)) / Float64(-x))
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]

5. Applied rewrites 24.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

7. Applied rewrites 69.5%

   \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{x} \cdot x, 0.3333333333333333, \mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{-x} \cdot \color{blue}{\frac{-1}{x}} \]

8. Taylor expanded in x around inf

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

10. Applied rewrites 96.7%

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

12. Applied rewrites 96.8%

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

13. Final simplification 96.8%

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

Alternative 3: 88.8% 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 7.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. 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 46.3

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

5. Applied rewrites 46.3%

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

7. Applied rewrites 88.5%

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

Alternative 4: 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 7.4%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow-to-exp N/A

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

   5. pow-exp N/A

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

   6. *-commutative N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. pow-to-exp N/A

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

   12. lift-cbrt.f64 N/A

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

   13. rem-cube-cbrt N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-log1p.f64 6.0

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

4. Applied rewrites 6.0%

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

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 2024283 
(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)))