2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.1%

Time: 7.8s

Alternatives: 8

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: 7.0% 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: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{\sqrt[3]{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/ x (* (cbrt x) (/ (fma 0.3333333333333333 x -0.1111111111111111) x)))
  -1.0))

C

double code(double x) {
	return pow((x / (cbrt(x) * (fma(0.3333333333333333, x, -0.1111111111111111) / x))), -1.0);
}

Julia

function code(x)
	return Float64(x / Float64(cbrt(x) * Float64(fma(0.3333333333333333, x, -0.1111111111111111) / x))) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(x / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}}{{x}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \color{blue}{\frac{-1}{9} \cdot \sqrt[3]{x}}\right)}{{x}^{2}} \]

   12. lower-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \color{blue}{\sqrt[3]{x}}\right)}{{x}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   14. lower-*.f64 23.9

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

5. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
6. Step-by-step derivation

7. Applied rewrites 75.8%

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

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

    \[\leadsto {\left(\frac{x}{\sqrt[3]{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}}\right)}^{-1} \]
11. Add Preprocessing

Alternative 2: 92.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (pow (* x x) -1.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt(pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt(Math.pow((x * x), -1.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt((Float64(x * x) ^ -1.0)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   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

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

      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 96.4

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

   5. Applied rewrites 96.4%

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

   7. Applied rewrites 96.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 6.2

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

   5. Applied rewrites 6.2%

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

   7. Applied rewrites 89.1%

      \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left({x}^{0.6666666666666666}\right)}^{-1} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (pow x 0.6666666666666666) -1.0) 0.3333333333333333))

C

double code(double x) {
	return pow(pow(x, 0.6666666666666666), -1.0) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x ** 0.6666666666666666d0) ** (-1.0d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(x, 0.6666666666666666), -1.0) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(math.pow(x, 0.6666666666666666), -1.0) * 0.3333333333333333

Julia

function code(x)
	return Float64(((x ^ 0.6666666666666666) ^ -1.0) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = ((x ^ 0.6666666666666666) ^ -1.0) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, 0.6666666666666666], $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.1

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

5. Applied rewrites 54.1%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]
8. Step-by-step derivation

9. Applied rewrites 89.5%

   \[\leadsto \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \]

10. Final simplification 89.5%

    \[\leadsto {\left({x}^{0.6666666666666666}\right)}^{-1} \cdot 0.3333333333333333 \]
11. Add Preprocessing

Alternative 4: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (cbrt x) (/ (fma 0.3333333333333333 x -0.1111111111111111) x)) x))

C

double code(double x) {
	return (cbrt(x) * (fma(0.3333333333333333, x, -0.1111111111111111) / x)) / x;
}

Julia

function code(x)
	return Float64(Float64(cbrt(x) * Float64(fma(0.3333333333333333, x, -0.1111111111111111) / x)) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}}{{x}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \color{blue}{\frac{-1}{9} \cdot \sqrt[3]{x}}\right)}{{x}^{2}} \]

   12. lower-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \color{blue}{\sqrt[3]{x}}\right)}{{x}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   14. lower-*.f64 23.9

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

5. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
6. Step-by-step derivation

7. Applied rewrites 75.8%

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

9. Applied rewrites 98.6%

   \[\leadsto \frac{\sqrt[3]{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}}{\color{blue}{x}} \]
10. Add Preprocessing

Alternative 5: 97.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.0%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}}{{x}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt[3]{{x}^{2} \cdot {x}^{2}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. pow-sqr N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{\left(2 \cdot 2\right)}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{\color{blue}{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{{x}^{4}}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \color{blue}{\frac{-1}{9} \cdot \sqrt[3]{x}}\right)}{{x}^{2}} \]

   12. lower-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \color{blue}{\sqrt[3]{x}}\right)}{{x}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{\color{blue}{x \cdot x}} \]

   14. lower-*.f64 23.9

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

5. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
6. Step-by-step derivation

7. Applied rewrites 75.8%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 97.8%

    \[\leadsto \frac{-1}{x} \cdot \left(-0.3333333333333333 \cdot \color{blue}{\sqrt[3]{x}}\right) \]
11. Add Preprocessing

Alternative 6: 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 6.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.1

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

5. Applied rewrites 54.1%

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

7. Applied rewrites 89.5%

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

Alternative 7: 5.3% 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(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 6.0%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 1.8

      \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]

7. Applied rewrites 1.8%

   \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-neg N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 5.4

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

9. Applied rewrites 5.4%

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

Alternative 8: 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 6.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024321 
(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)))