2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.7%

Time: 3.5s

Alternatives: 12

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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\left(x - -1\right) \cdot \left(x - -1\right)} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+152)
   (/
    1.0
    (+
     (cbrt (* (- x -1.0) (- x -1.0)))
     (+ (cbrt (* x x)) (cbrt (* (- x -1.0) x)))))
   (* (/ (cbrt (* (/ 1.0 x) 1.0)) (cbrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+152) {
		tmp = 1.0 / (cbrt(((x - -1.0) * (x - -1.0))) + (cbrt((x * x)) + cbrt(((x - -1.0) * x))));
	} else {
		tmp = (cbrt(((1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+152) {
		tmp = 1.0 / (Math.cbrt(((x - -1.0) * (x - -1.0))) + (Math.cbrt((x * x)) + Math.cbrt(((x - -1.0) * x))));
	} else {
		tmp = (Math.cbrt(((1.0 / x) * 1.0)) / Math.cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+152)
		tmp = Float64(1.0 / Float64(cbrt(Float64(Float64(x - -1.0) * Float64(x - -1.0))) + Float64(cbrt(Float64(x * x)) + cbrt(Float64(Float64(x - -1.0) * x)))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+152], N[(1.0 / N[(N[Power[N[(N[(x - -1.0), $MachinePrecision] * N[(x - -1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] * 1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e152

   1. Initial program 9.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 13.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.1%

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

   if 2.0000000000000001e152 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. pow1/3 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 5.8

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

   6. Applied rewrites 5.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. frac-times N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 8.5

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

   8. Applied rewrites 8.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. cbrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

       \[\leadsto \frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.5e+14)
   (/
    (- (- x -1.0) x)
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (* (- x -1.0) x)))))
   (* (/ (cbrt (* (/ 1.0 x) 1.0)) (cbrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 4.5e+14) {
		tmp = ((x - -1.0) - x) / (pow((x - -1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt(((x - -1.0) * x))));
	} else {
		tmp = (cbrt(((1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.5e+14) {
		tmp = ((x - -1.0) - x) / (Math.pow((x - -1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) + Math.cbrt(((x - -1.0) * x))));
	} else {
		tmp = (Math.cbrt(((1.0 / x) * 1.0)) / Math.cbrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.5e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(Float64(Float64(x - -1.0) * x)))));
	else
		tmp = Float64(Float64(cbrt(Float64(Float64(1.0 / x) * 1.0)) / cbrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.5e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x - -1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] * 1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5e14

   1. Initial program 59.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 99.1%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. unpow-prod-down N/A

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

      7. pow2 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 98.3

          \[\leadsto \frac{\left(x - -1\right) - x}{\color{blue}{{\left(x - -1\right)}^{0.6666666666666666}} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 97.3

          \[\leadsto \frac{\left(x - -1\right) - x}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\color{blue}{{x}^{0.6666666666666666}} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(x - -1\right) - x}{\color{blue}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)}} \]

   if 4.5e14 < x

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. pow1/3 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 49.3

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

   6. Applied rewrites 49.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. frac-times N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 50.7

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

   8. Applied rewrites 50.7%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. cbrt-div N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

       \[\leadsto \frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (fma
   (pow x -2.6666666666666665)
   0.06172839506172839
   (* (/ (/ 1.0 (cbrt x)) (cbrt x)) 0.3333333333333333))))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, fma(pow(x, -2.6666666666666665), 0.06172839506172839, (((1.0 / cbrt(x)) / cbrt(x)) * 0.3333333333333333)));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, fma((x ^ -2.6666666666666665), 0.06172839506172839, Float64(Float64(Float64(1.0 / cbrt(x)) / cbrt(x)) * 0.3333333333333333)))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -2.6666666666666665], $MachinePrecision] * 0.06172839506172839 + N[(N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 45.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.06172839506172839, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \color{blue}{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(5\right)\right)\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(5\right)\right)\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{8}}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right) \]

7. Applied rewrites 90.1%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right)\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right)\right) \]

   5. pow-flip N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right)\right) \]

   6. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{3}\right)\right) \]

   9. cbrt-div N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   11. lower-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   13. lift-cbrt.f64 97.8

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]

9. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

    2. lift-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

    3. cbrt-div N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

    5. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

    6. lift-cbrt.f64 97.8

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]

11. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]
12. Add Preprocessing

Alternative 4: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (fma
   (pow x -2.6666666666666665)
   0.06172839506172839
   (* (/ (cbrt (/ 1.0 x)) (cbrt x)) 0.3333333333333333))))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, fma(pow(x, -2.6666666666666665), 0.06172839506172839, ((cbrt((1.0 / x)) / cbrt(x)) * 0.3333333333333333)));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, fma((x ^ -2.6666666666666665), 0.06172839506172839, Float64(Float64(cbrt(Float64(1.0 / x)) / cbrt(x)) * 0.3333333333333333)))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -2.6666666666666665], $MachinePrecision] * 0.06172839506172839 + N[(N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 45.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.06172839506172839, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \color{blue}{\left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \frac{-1}{9} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \color{blue}{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{5}}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{fma}\left({\left({x}^{\left(\mathsf{neg}\left(5\right)\right)}\right)}^{\frac{1}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\left(\mathsf{neg}\left(5\right)\right) \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(-5 \cdot \frac{1}{3}\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(5\right)\right)\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   10. lower-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(5\right)\right)\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -5\right)}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left(\sqrt[3]{\frac{1}{{x}^{8}}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right) \]

7. Applied rewrites 90.1%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\frac{-2}{3}} \cdot \frac{1}{3}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\left(-2 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{1}{3}\right)} \cdot \frac{1}{3}\right)\right) \]

   4. pow-pow N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {\left({x}^{\left(\mathsf{neg}\left(2\right)\right)}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right)\right) \]

   5. pow-flip N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} \cdot \frac{1}{3}\right)\right) \]

   6. pow1/3 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{1}{3}\right)\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot \frac{1}{3}\right)\right) \]

   9. cbrt-div N/A

      \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   11. lower-cbrt.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{-1}{9}, \mathsf{fma}\left({x}^{\frac{-8}{3}}, \frac{5}{81}, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot \frac{1}{3}\right)\right) \]

   13. lift-cbrt.f64 97.8

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]

9. Applied rewrites 97.8%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, \mathsf{fma}\left({x}^{-2.6666666666666665}, 0.06172839506172839, \frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{x}} \cdot 0.3333333333333333\right)\right) \]
10. Add Preprocessing

Alternative 5: 96.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.8

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

4. Applied rewrites 88.8%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. pow2 N/A

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

   8. frac-times N/A

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

   9. pow1/3 N/A

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

   10. lower-cbrt.f64 N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. pow2 N/A

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

   14. lower-/.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 49.8

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

6. Applied rewrites 49.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. frac-times N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 51.2

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

8. Applied rewrites 51.2%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-*r/ N/A

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

   6. cbrt-div N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-cbrt.f64 96.5

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

10. Applied rewrites 96.5%

    \[\leadsto \frac{\sqrt[3]{\frac{1}{x} \cdot 1}}{\sqrt[3]{x}} \cdot 0.3333333333333333 \]
11. Add Preprocessing

Alternative 6: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\log x}\right)}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (pow (exp (log x)) -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = pow(exp(log(x)), -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.pow(Math.exp(Math.log(x)), -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64((exp(log(x)) ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. cbrt-prod N/A

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

      13. pow2 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. pow1/3 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 4.7

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

   6. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. frac-times N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 7.5

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

   8. Applied rewrites 7.5%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow1/3 N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. pow-flip N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-to-exp N/A

          \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      14. exp-prod N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-exp.f64 N/A

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

      17. lower-log.f64 89.8

          \[\leadsto {\left(e^{\log x}\right)}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

   10. Applied rewrites 89.8%

       \[\leadsto {\left(e^{\log x}\right)}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. cbrt-prod N/A

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

      13. pow2 N/A

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

      14. lower-cbrt.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      3. lower-exp.f64 N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      5. lift-log.f64 89.4

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

   6. Applied rewrites 89.4%

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

Alternative 8: 92.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. pow2 N/A

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

      8. frac-times N/A

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

      9. pow1/3 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. pow2 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      3. lower-exp.f64 N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      5. lift-log.f64 89.4

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

   6. Applied rewrites 89.4%

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

Alternative 9: 89.1% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333))

C

double code(double x) {
	return exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x):
	return math.exp((math.log(x) * -0.6666666666666666)) * 0.3333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.8

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

4. Applied rewrites 88.8%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

      \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

   3. lower-exp.f64 N/A

      \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

   4. lower-*.f64 N/A

      \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

   5. lift-log.f64 89.1

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

6. Applied rewrites 89.1%

   \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]
7. Add Preprocessing

Alternative 10: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
5. Add Preprocessing

Alternative 11: 1.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ -\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (expm1 (* (log x) 0.3333333333333333))))

C

double code(double x) {
	return -expm1((log(x) * 0.3333333333333333));
}

Java

public static double code(double x) {
	return -Math.expm1((Math.log(x) * 0.3333333333333333));
}

Python

def code(x):
	return -math.expm1((math.log(x) * 0.3333333333333333))

Julia

function code(x)
	return Float64(-expm1(Float64(log(x) * 0.3333333333333333)))
end

Wolfram

code[x_] := (-N[(Exp[N[(N[Log[x], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]] - 1), $MachinePrecision])

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

   1. negate-sub2 N/A

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

   2. lower-neg.f64 N/A

      \[\leadsto -\left(\sqrt[3]{x} - 1\right) \]

   3. pow1/3 N/A

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

   4. pow-to-exp N/A

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

   5. lower-expm1.f64 N/A

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot \frac{1}{3}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot \frac{1}{3}\right) \]

   7. lower-log.f64 1.8

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right) \]

4. Applied rewrites 1.8%

   \[\leadsto \color{blue}{-\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right)} \]
5. Add Preprocessing

Alternative 12: 1.8% accurate, 2.1× 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 Float64(-cbrt(x))
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

   1. negate-sub2 N/A

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

   2. lower-neg.f64 N/A

      \[\leadsto -\left(\sqrt[3]{x} - 1\right) \]

   3. pow1/3 N/A

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

   4. pow-to-exp N/A

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

   5. lower-expm1.f64 N/A

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot \frac{1}{3}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot \frac{1}{3}\right) \]

   7. lower-log.f64 1.8

      \[\leadsto -\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right) \]

4. Applied rewrites 1.8%

   \[\leadsto \color{blue}{-\mathsf{expm1}\left(\log x \cdot 0.3333333333333333\right)} \]

5. Taylor expanded in x around inf

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

   1. lift-cbrt.f64 1.8

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

7. Applied rewrites 1.8%

   \[\leadsto -\sqrt[3]{x} \]
8. 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 2025119 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform c (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

  (- (cbrt (+ x 1.0)) (cbrt x)))