2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 90.6%

Time: 3.8s

Alternatives: 15

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: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 17000000:\\ \;\;\;\;\frac{1}{{\left(x - -1\right)}^{-0.3333333333333333}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 17000000.0)
   (- (/ 1.0 (pow (- x -1.0) -0.3333333333333333)) (cbrt x))
   (exp (* (log (* (pow x 0.6666666666666666) 3.0)) -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = (1.0 / pow((x - -1.0), -0.3333333333333333)) - cbrt(x);
	} else {
		tmp = exp((log((pow(x, 0.6666666666666666) * 3.0)) * -1.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 17000000.0) {
		tmp = (1.0 / Math.pow((x - -1.0), -0.3333333333333333)) - Math.cbrt(x);
	} else {
		tmp = Math.exp((Math.log((Math.pow(x, 0.6666666666666666) * 3.0)) * -1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 17000000.0)
		tmp = Float64(Float64(1.0 / (Float64(x - -1.0) ^ -0.3333333333333333)) - cbrt(x));
	else
		tmp = exp(Float64(log(Float64((x ^ 0.6666666666666666) * 3.0)) * -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 17000000.0], N[(N[(1.0 / N[Power[N[(x - -1.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[N[(N[Power[x, 0.6666666666666666], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e7

   1. Initial program 7.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift--.f64 N/A

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

      4. metadata-eval N/A

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

      5. add-flip N/A

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

      6. add-flip N/A

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

      7. metadata-eval N/A

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

      8. lift--.f64 N/A

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

      9. remove-double-neg N/A

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

      10. pow-neg N/A

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

      11. lower-unsound-/.f64 N/A

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

      12. lower-unsound-pow.f64 N/A

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

      13. metadata-eval 4.6

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

   5. Applied rewrites 4.6%

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

   if 1.7e7 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lift-pow.f64 N/A

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

      7. mult-flip N/A

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

      8. div-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-to-exp N/A

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

      12. lower-unsound-exp.f64 N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-unsound-log.f64 89.5

          \[\leadsto e^{\log \left(\frac{{x}^{0.6666666666666666}}{0.3333333333333333}\right) \cdot -1} \]

      15. lift-/.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 89.5

          \[\leadsto e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1} \]

   8. Applied rewrites 89.5%

      \[\leadsto e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 35000000:\\ \;\;\;\;\sqrt[3]{x - -1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 35000000.0)
   (- (cbrt (- x -1.0)) (cbrt x))
   (exp (* (log (* (pow x 0.6666666666666666) 3.0)) -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 35000000.0) {
		tmp = cbrt((x - -1.0)) - cbrt(x);
	} else {
		tmp = exp((log((pow(x, 0.6666666666666666) * 3.0)) * -1.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 35000000.0) {
		tmp = Math.cbrt((x - -1.0)) - Math.cbrt(x);
	} else {
		tmp = Math.exp((Math.log((Math.pow(x, 0.6666666666666666) * 3.0)) * -1.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 35000000.0)
		tmp = Float64(cbrt(Float64(x - -1.0)) - cbrt(x));
	else
		tmp = exp(Float64(log(Float64((x ^ 0.6666666666666666) * 3.0)) * -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 35000000.0], N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[N[(N[Power[x, 0.6666666666666666], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e7

   1. Initial program 7.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

   if 3.5e7 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. pow-flip N/A

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

      6. lift-pow.f64 N/A

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

      7. mult-flip N/A

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

      8. div-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-to-exp N/A

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

      12. lower-unsound-exp.f64 N/A

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

      13. lower-unsound-*.f64 N/A

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

      14. lower-unsound-log.f64 89.5

          \[\leadsto e^{\log \left(\frac{{x}^{0.6666666666666666}}{0.3333333333333333}\right) \cdot -1} \]

      15. lift-/.f64 N/A

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

      16. mult-flip N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 89.5

          \[\leadsto e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1} \]

   8. Applied rewrites 89.5%

      \[\leadsto e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;e^{\log \left({x}^{0.6666666666666666}\right) \cdot -1} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x - -1} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-6)
   (* (exp (* (log (pow x 0.6666666666666666)) -1.0)) 0.3333333333333333)
   (- (cbrt (- x -1.0)) (cbrt x))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 5e-6) {
		tmp = exp((log(pow(x, 0.6666666666666666)) * -1.0)) * 0.3333333333333333;
	} else {
		tmp = cbrt((x - -1.0)) - cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.cbrt((x + 1.0)) - Math.cbrt(x)) <= 5e-6) {
		tmp = Math.exp((Math.log(Math.pow(x, 0.6666666666666666)) * -1.0)) * 0.3333333333333333;
	} else {
		tmp = Math.cbrt((x - -1.0)) - Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 5e-6)
		tmp = Float64(exp(Float64(log((x ^ 0.6666666666666666)) * -1.0)) * 0.3333333333333333);
	else
		tmp = Float64(cbrt(Float64(x - -1.0)) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[Exp[N[(N[Log[N[Power[x, 0.6666666666666666], $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000041e-6

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lift-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. pow-to-exp N/A

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

      7. lower-unsound-exp.f64 N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-log.f64 89.5

         \[\leadsto e^{\log \left({x}^{0.6666666666666666}\right) \cdot -1} \cdot 0.3333333333333333 \]

   8. Applied rewrites 89.5%

      \[\leadsto e^{\log \left({x}^{0.6666666666666666}\right) \cdot -1} \cdot 0.3333333333333333 \]

   if 5.00000000000000041e-6 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 7.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\sqrt[3]{x - -1} - \sqrt[3]{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{4}{{x}^{0.6666666666666666}}\right) + \log 0.08333333333333333}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  -0.1111111111111111
  (pow x -1.6666666666666667)
  (exp
   (+ (log (/ 4.0 (pow x 0.6666666666666666))) (log 0.08333333333333333)))))

C

double code(double x) {
	return fma(-0.1111111111111111, pow(x, -1.6666666666666667), exp((log((4.0 / pow(x, 0.6666666666666666))) + log(0.08333333333333333))));
}

Julia

function code(x)
	return fma(-0.1111111111111111, (x ^ -1.6666666666666667), exp(Float64(log(Float64(4.0 / (x ^ 0.6666666666666666))) + log(0.08333333333333333))))
end

Wolfram

code[x_] := N[(-0.1111111111111111 * N[Power[x, -1.6666666666666667], $MachinePrecision] + N[Exp[N[(N[Log[N[(4.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[0.08333333333333333], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]

6. Applied rewrites 89.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. lift-pow.f64 N/A

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

   7. mult-flip N/A

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

   8. div-flip-rev N/A

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

   9. lift-/.f64 N/A

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

   10. inv-pow N/A

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

   11. pow-to-exp N/A

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

   12. lower-unsound-exp.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-unsound-log.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{{x}^{0.6666666666666666}}{0.3333333333333333}\right) \cdot -1}\right) \]

   15. lift-/.f64 N/A

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

   16. mult-flip N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]

8. Applied rewrites 90.4%

   \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f64 N/A

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

   4. log-pow-rev N/A

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

   5. unpow-1 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4 \cdot \frac{1}{4}}{{x}^{\frac{2}{3}} \cdot 3}\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4 \cdot \frac{1}{4}}{{x}^{\frac{2}{3}} \cdot 3}\right)}\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}} \cdot \frac{\frac{1}{4}}{3}\right)}\right) \]

   9. log-prod N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}}\right) + \log \left(\frac{\frac{1}{4}}{3}\right)}\right) \]

   10. lower-unsound-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}}\right) + \log \left(\frac{\frac{1}{4}}{3}\right)}\right) \]

   11. lower-unsound-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}}\right) + \log \left(\frac{\frac{1}{4}}{3}\right)}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}}\right) + \log \left(\frac{\frac{1}{4}}{3}\right)}\right) \]

   13. lower-unsound-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{4}{{x}^{\frac{2}{3}}}\right) + \log \left(\frac{\frac{1}{4}}{3}\right)}\right) \]

   14. metadata-eval 90.6

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{4}{{x}^{0.6666666666666666}}\right) + \log 0.08333333333333333}\right) \]

10. Applied rewrites 90.6%

    \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{4}{{x}^{0.6666666666666666}}\right) + \log 0.08333333333333333}\right) \]
11. Add Preprocessing

Alternative 5: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x - -1} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 5e-6)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)
   (- (cbrt (- x -1.0)) (cbrt x))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 5e-6) {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	} else {
		tmp = cbrt((x - -1.0)) - cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((Math.cbrt((x + 1.0)) - Math.cbrt(x)) <= 5e-6) {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	} else {
		tmp = Math.cbrt((x - -1.0)) - Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 5e-6)
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	else
		tmp = Float64(cbrt(Float64(x - -1.0)) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000041e-6

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
   7. 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-unsound-exp.f64 N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-log.f64 89.1

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

   8. Applied rewrites 89.1%

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

   if 5.00000000000000041e-6 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 7.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

      \[\leadsto \color{blue}{\sqrt[3]{x - -1} - \sqrt[3]{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  -0.1111111111111111
  (pow x -1.6666666666666667)
  (exp (* (log (* (pow x 0.6666666666666666) 3.0)) -1.0))))

C

double code(double x) {
	return fma(-0.1111111111111111, pow(x, -1.6666666666666667), exp((log((pow(x, 0.6666666666666666) * 3.0)) * -1.0)));
}

Julia

function code(x)
	return fma(-0.1111111111111111, (x ^ -1.6666666666666667), exp(Float64(log(Float64((x ^ 0.6666666666666666) * 3.0)) * -1.0)))
end

Wolfram

code[x_] := N[(-0.1111111111111111 * N[Power[x, -1.6666666666666667], $MachinePrecision] + N[Exp[N[(N[Log[N[(N[Power[x, 0.6666666666666666], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]

6. Applied rewrites 89.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. lift-pow.f64 N/A

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

   7. mult-flip N/A

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

   8. div-flip-rev N/A

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

   9. lift-/.f64 N/A

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

   10. inv-pow N/A

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

   11. pow-to-exp N/A

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

   12. lower-unsound-exp.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-unsound-log.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{{x}^{0.6666666666666666}}{0.3333333333333333}\right) \cdot -1}\right) \]

   15. lift-/.f64 N/A

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

   16. mult-flip N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]

8. Applied rewrites 90.4%

   \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]
9. Add Preprocessing

Alternative 7: 90.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  -0.1111111111111111
  (pow x -1.6666666666666667)
  (exp (log (* (pow x -0.6666666666666666) 0.3333333333333333)))))

C

double code(double x) {
	return fma(-0.1111111111111111, pow(x, -1.6666666666666667), exp(log((pow(x, -0.6666666666666666) * 0.3333333333333333))));
}

Julia

function code(x)
	return fma(-0.1111111111111111, (x ^ -1.6666666666666667), exp(log(Float64((x ^ -0.6666666666666666) * 0.3333333333333333))))
end

Wolfram

code[x_] := N[(-0.1111111111111111 * N[Power[x, -1.6666666666666667], $MachinePrecision] + N[Exp[N[Log[N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]

6. Applied rewrites 89.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. lift-pow.f64 N/A

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

   7. mult-flip N/A

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

   8. div-flip-rev N/A

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

   9. lift-/.f64 N/A

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

   10. inv-pow N/A

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

   11. pow-to-exp N/A

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

   12. lower-unsound-exp.f64 N/A

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

   13. lower-unsound-*.f64 N/A

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

   14. lower-unsound-log.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(\frac{{x}^{0.6666666666666666}}{0.3333333333333333}\right) \cdot -1}\right) \]

   15. lift-/.f64 N/A

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

   16. mult-flip N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 90.4

       \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]

8. Applied rewrites 90.4%

   \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right) \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-log.f64 N/A

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

   4. log-pow-rev N/A

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

   5. exp-to-pow N/A

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

   6. lift-log.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lower-log.f64 90.4

      \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left(e^{\log \left({x}^{0.6666666666666666} \cdot 3\right) \cdot -1}\right)}\right) \]

   10. lift-exp.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. exp-to-pow N/A

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

   14. unpow-1 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-/r* N/A

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

   17. lift-pow.f64 N/A

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

   18. pow-flip N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{-1}{9}, {x}^{\frac{-5}{3}}, e^{\log \left(\frac{{x}^{\left(\frac{-5}{3} + 1\right)}}{3}\right)}\right) \]

   21. pow-plus N/A

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

   22. lift-pow.f64 N/A

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

   23. mult-flip-rev N/A

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

   24. metadata-eval N/A

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

10. Applied rewrites 90.4%

    \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, e^{\log \left({x}^{-0.6666666666666666} \cdot 0.3333333333333333\right)}\right) \]
11. Add Preprocessing

Alternative 8: 90.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(e^{\log x \cdot -0.6666666666666666}, 0.3333333333333333, -0.1111111111111111 \cdot {x}^{-1.6666666666666667}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (exp (* (log x) -0.6666666666666666))
  0.3333333333333333
  (* -0.1111111111111111 (pow x -1.6666666666666667))))

C

double code(double x) {
	return fma(exp((log(x) * -0.6666666666666666)), 0.3333333333333333, (-0.1111111111111111 * pow(x, -1.6666666666666667)));
}

Julia

function code(x)
	return fma(exp(Float64(log(x) * -0.6666666666666666)), 0.3333333333333333, Float64(-0.1111111111111111 * (x ^ -1.6666666666666667)))
end

Wolfram

code[x_] := N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, -1.6666666666666667], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

      \[\leadsto x \cdot \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)}\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}}\right)\right) \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   9. pow-flip N/A

      \[\leadsto \frac{1}{3} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   10. pow-plus N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{5}{3}\right)\right) + 1\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   14. pow-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   17. lower-fma.f64 N/A

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

6. Applied rewrites 89.7%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. lower-unsound-exp.f64 N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound-log.f64 90.0

      \[\leadsto \mathsf{fma}\left(e^{\log x \cdot -0.6666666666666666}, 0.3333333333333333, -0.1111111111111111 \cdot {x}^{-1.6666666666666667}\right) \]

8. Applied rewrites 90.0%

   \[\leadsto \mathsf{fma}\left(e^{\log x \cdot -0.6666666666666666}, 0.3333333333333333, -0.1111111111111111 \cdot {x}^{-1.6666666666666667}\right) \]
9. Add Preprocessing

Alternative 9: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, \frac{0.3333333333333333}{{x}^{0.6666666666666666}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  -0.1111111111111111
  (pow x -1.6666666666666667)
  (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	return fma(-0.1111111111111111, pow(x, -1.6666666666666667), (0.3333333333333333 / pow(x, 0.6666666666666666)));
}

Julia

function code(x)
	return fma(-0.1111111111111111, (x ^ -1.6666666666666667), Float64(0.3333333333333333 / (x ^ 0.6666666666666666)))
end

Wolfram

code[x_] := N[(-0.1111111111111111 * N[Power[x, -1.6666666666666667], $MachinePrecision] + N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]

6. Applied rewrites 89.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. metadata-eval N/A

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

   5. pow-flip N/A

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

   6. lift-pow.f64 N/A

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

   7. mult-flip N/A

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

   8. lower-/.f64 89.7

      \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, \frac{0.3333333333333333}{{x}^{0.6666666666666666}}\right) \]

8. Applied rewrites 89.7%

   \[\leadsto \mathsf{fma}\left(-0.1111111111111111, {x}^{-1.6666666666666667}, \frac{0.3333333333333333}{{x}^{0.6666666666666666}}\right) \]
9. Add Preprocessing

Alternative 10: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot -0.1111111111111111\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -0.6666666666666666)
  0.3333333333333333
  (* (pow x -1.6666666666666667) -0.1111111111111111)))

C

double code(double x) {
	return fma(pow(x, -0.6666666666666666), 0.3333333333333333, (pow(x, -1.6666666666666667) * -0.1111111111111111));
}

Julia

function code(x)
	return fma((x ^ -0.6666666666666666), 0.3333333333333333, Float64((x ^ -1.6666666666666667) * -0.1111111111111111))
end

Wolfram

code[x_] := N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

      \[\leadsto x \cdot \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)}\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}}\right)\right) \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   9. pow-flip N/A

      \[\leadsto \frac{1}{3} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   10. pow-plus N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{5}{3}\right)\right) + 1\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   14. pow-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   17. lower-fma.f64 N/A

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

6. Applied rewrites 89.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 89.7

      \[\leadsto \mathsf{fma}\left({x}^{-0.6666666666666666}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot -0.1111111111111111\right) \]

8. Applied rewrites 89.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.3333333333333333, {x}^{-1.6666666666666667} \cdot -0.1111111111111111\right)} \]
9. Add Preprocessing

Alternative 11: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(x, -0.6666666666666666) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((x ^ -0.6666666666666666) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 55.5

      \[\leadsto x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{\color{blue}{2.6666666666666665}}}\right) \]

4. Applied rewrites 55.5%

   \[\leadsto \color{blue}{x \cdot \left(0.3333333333333333 \cdot \frac{1}{{x}^{1.6666666666666667}} - 0.1111111111111111 \cdot \frac{1}{{x}^{2.6666666666666665}}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-flip N/A

      \[\leadsto x \cdot \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)}\right) \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x} \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{{x}^{\frac{5}{3}}}\right) \cdot x + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}}\right)\right) \cdot x \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   7. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   8. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{{x}^{\frac{5}{3}}} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   9. pow-flip N/A

      \[\leadsto \frac{1}{3} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   10. pow-plus N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{5}{3}\right)\right) + 1\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{1}{3} \cdot {x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   14. pow-flip N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   15. lift-pow.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{1}{{x}^{\frac{2}{3}}} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right) \cdot x \]

   16. *-commutative N/A

       \[\leadsto \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{8}{3}}}\right)\right)} \cdot x \]

   17. lower-fma.f64 N/A

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

6. Applied rewrites 89.7%

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

   1. lift-fma.f64 N/A

      \[\leadsto {x}^{\frac{-2}{3}} \cdot \frac{1}{3} + \color{blue}{\frac{-1}{9} \cdot {x}^{\frac{-5}{3}}} \]

   2. lift-*.f64 N/A

      \[\leadsto {x}^{\frac{-2}{3}} \cdot \frac{1}{3} + \color{blue}{\frac{-1}{9}} \cdot {x}^{\frac{-5}{3}} \]

   3. +-commutative N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} + \color{blue}{{x}^{\frac{-2}{3}} \cdot \frac{1}{3}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} + \color{blue}{{x}^{\frac{-2}{3}}} \cdot \frac{1}{3} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} + \color{blue}{{x}^{\frac{-2}{3}}} \cdot \frac{1}{3} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} + {x}^{\frac{-2}{3}} \cdot \color{blue}{\frac{1}{3}} \]

   7. *-commutative N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} + \frac{1}{3} \cdot \color{blue}{{x}^{\frac{-2}{3}}} \]

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

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot {x}^{\frac{-2}{3}}} \]

   9. *-commutative N/A

      \[\leadsto \frac{-1}{9} \cdot {x}^{\frac{-5}{3}} - {x}^{\frac{-2}{3}} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \]

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

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-lft-neg-in N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. lift-*.f64 N/A

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

   16. remove-double-neg N/A

       \[\leadsto {x}^{\frac{-5}{3}} \cdot \frac{-1}{9} + {x}^{\frac{-2}{3}} \cdot \color{blue}{\frac{1}{3}} \]

   17. lower-fma.f64 89.7

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]

8. Applied rewrites 89.7%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
9. Add Preprocessing

Alternative 12: 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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.8

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

6. Applied rewrites 88.8%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
7. 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-unsound-exp.f64 N/A

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

   4. lower-unsound-*.f64 N/A

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

   5. lower-unsound-log.f64 89.1

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

8. Applied rewrites 89.1%

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

Alternative 13: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 / pow(x, 0.6666666666666666);
}

Fortran

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 = 0.3333333333333333d0 / (x ** 0.6666666666666666d0)
end function

Java

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

Python

def code(x):
	return 0.3333333333333333 / math.pow(x, 0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 / (x ^ 0.6666666666666666);
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

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

Alternative 14: 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

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.8

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

6. Applied rewrites 88.8%

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

Alternative 15: 4.1% accurate, 36.6× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 4.1%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{\left(t\_0 \cdot t\_0 + \sqrt[3]{x} \cdot t\_0\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (/ 1.0 (+ (+ (* t_0 t_0) (* (cbrt x) t_0)) (* (cbrt x) (cbrt x))))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (cbrt(x) * t_0)) + (cbrt(x) * cbrt(x)));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (Math.cbrt(x) * t_0)) + (Math.cbrt(x) * Math.cbrt(x)));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) + Float64(cbrt(x) * t_0)) + Float64(cbrt(x) * cbrt(x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025154 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  :herbie-expected 5/2

  :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)))