2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 99.0%

Time: 9.0s

Alternatives: 8

Speedup: 1.9×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))

C

double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

Java

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

Julia

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{x + 1} - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))

C

double code(double x) {
	return cbrt((x + 1.0)) - cbrt(x);
}

Java

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

Julia

function code(x)
	return Float64(cbrt(Float64(x + 1.0)) - cbrt(x))
end

Wolfram

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

Alternative 1: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{x} + {\left(\sqrt[3]{x}\right)}^{2}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9e+14)
   (/
    (- (+ 1.0 x) x)
    (+
     (+ (* (cbrt (+ 1.0 x)) (cbrt x)) (pow (cbrt x) 2.0))
     (exp (* 0.6666666666666666 (log1p x)))))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9e+14) {
		tmp = ((1.0 + x) - x) / (((cbrt((1.0 + x)) * cbrt(x)) + pow(cbrt(x), 2.0)) + exp((0.6666666666666666 * log1p(x))));
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.9e+14) {
		tmp = ((1.0 + x) - x) / (((Math.cbrt((1.0 + x)) * Math.cbrt(x)) + Math.pow(Math.cbrt(x), 2.0)) + Math.exp((0.6666666666666666 * Math.log1p(x))));
	} else {
		tmp = 0.3333333333333333 / (x / Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(Float64(Float64(cbrt(Float64(1.0 + x)) * cbrt(x)) + (cbrt(x) ^ 2.0)) + exp(Float64(0.6666666666666666 * log1p(x)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.9e+14], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e14

   1. Initial program 54.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 50.7

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 50.7%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{\frac{1}{6}}\right)}^{2}} \]

      2. unpow2 N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - {x}^{\frac{1}{6}} \cdot \color{blue}{{x}^{\frac{1}{6}}} \]

      4. sqr-pow N/A

         \[\leadsto \sqrt[3]{x + 1} - {x}^{\frac{1}{6}} \cdot \color{blue}{\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}} \]

      7. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - \left(\color{blue}{{x}^{\frac{1}{6}}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      8. pow-prod-up N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\left(\frac{1}{6} + \frac{\frac{1}{6}}{2}\right)}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \sqrt[3]{x + 1} - {x}^{\left(\frac{1}{6} + \color{blue}{\frac{1}{12}}\right)} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\color{blue}{\frac{1}{4}}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\color{blue}{\left(\frac{1}{12} \cdot 3\right)}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\left(\color{blue}{\frac{\frac{1}{6}}{2}} \cdot 3\right)} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      13. lower-pow.f64 N/A

          \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\left(\frac{\frac{1}{6}}{2} \cdot 3\right)}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\left(\color{blue}{\frac{1}{12}} \cdot 3\right)} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\color{blue}{\frac{1}{4}}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)} \]

      16. lower-pow.f64 N/A

          \[\leadsto \sqrt[3]{x + 1} - {x}^{\frac{1}{4}} \cdot \color{blue}{{x}^{\left(\frac{\frac{1}{6}}{2}\right)}} \]

      17. metadata-eval 53.5

          \[\leadsto \sqrt[3]{x + 1} - {x}^{0.25} \cdot {x}^{\color{blue}{0.08333333333333333}} \]

   6. Applied rewrites 53.5%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.25} \cdot {x}^{0.08333333333333333}} \]

   8. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \left({\left(\sqrt[3]{x}\right)}^{2} - \sqrt[3]{1 + x} \cdot \left(-\sqrt[3]{x}\right)\right)}} \]

   if 2.9e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 90.3%

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

   11. Applied rewrites 99.0%

       \[\leadsto \frac{0.3333333333333333}{\frac{x}{\color{blue}{\sqrt[3]{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{x} + {\left(\sqrt[3]{x}\right)}^{2}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.9e+14)
   (/
    (- (+ 1.0 x) x)
    (fma
     (cbrt x)
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (exp (* 0.6666666666666666 (log1p x)))))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.9e+14) {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), exp((0.6666666666666666 * log1p(x))));
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.9e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), exp(Float64(0.6666666666666666 * log1p(x)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.9e+14], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e14

   1. Initial program 54.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 50.7

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 50.7%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\sqrt[3]{x + 1} - {\left({x}^{\frac{1}{6}}\right)}^{2}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{\frac{1}{6}}\right)}^{2}} \]

      3. lift-pow.f64 N/A

         \[\leadsto \sqrt[3]{x + 1} - {\color{blue}{\left({x}^{\frac{1}{6}}\right)}}^{2} \]

      4. pow-pow N/A

         \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{\left(\frac{1}{6} \cdot 2\right)}} \]

      5. metadata-eval N/A

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

      6. pow1/3 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 97.7%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]

   if 2.9e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 90.3%

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

   11. Applied rewrites 99.0%

       \[\leadsto \frac{0.3333333333333333}{\frac{x}{\color{blue}{\sqrt[3]{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+59)
   (/
    (fma
     (cbrt (pow x 4.0))
     0.3333333333333333
     (fma
      (cbrt (/ 1.0 (* x x)))
      0.06172839506172839
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+59) {
		tmp = fma(cbrt(pow(x, 4.0)), 0.3333333333333333, fma(cbrt((1.0 / (x * x))), 0.06172839506172839, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+59)
		tmp = Float64(fma(cbrt((x ^ 4.0)), 0.3333333333333333, fma(cbrt(Float64(1.0 / Float64(x * x))), 0.06172839506172839, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+59], N[(N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e59

   1. Initial program 18.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 21.7

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied rewrites 21.7%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 97.9%

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

   if 4.9999999999999997e59 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 46.0

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 89.9%

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

   11. Applied rewrites 99.0%

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

Alternative 4: 92.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 94.9

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

   5. Applied rewrites 94.9%

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

   7. Applied rewrites 94.7%

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

   9. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 7.2

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

   5. Applied rewrites 7.2%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 89.0%

      \[\leadsto \frac{0.3333333333333333}{{x}^{\color{blue}{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 54.5

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

5. Applied rewrites 54.5%

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

7. Applied rewrites 96.4%

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

9. Applied rewrites 88.9%

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

11. Applied rewrites 97.0%

    \[\leadsto \frac{0.3333333333333333}{\frac{x}{\color{blue}{\sqrt[3]{x}}}} \]
12. Add Preprocessing

Alternative 6: 88.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 54.5

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

5. Applied rewrites 54.5%

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

7. Applied rewrites 88.9%

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

Alternative 7: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ 0.3333333333333333
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, 0.3333333333333333], $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. clear-num N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. rem-cube-cbrt N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow-to-exp N/A

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

   15. rec-exp N/A

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

   16. rem-log-exp N/A

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

   17. pow-to-exp N/A

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

4. Applied rewrites 6.3%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{-\mathsf{log1p}\left(x\right)}}}} - \sqrt[3]{x} \]

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-cbrt.f64 1.8

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 5.4%

   \[\leadsto {x}^{\color{blue}{0.3333333333333333}} \]
10. Add Preprocessing

Alternative 8: 5.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (cbrt x))

C

double code(double x) {
	return cbrt(x);
}

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. clear-num N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. rem-cube-cbrt N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow-to-exp N/A

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

   15. rec-exp N/A

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

   16. rem-log-exp N/A

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

   17. pow-to-exp N/A

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

4. Applied rewrites 6.3%

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{e^{-\mathsf{log1p}\left(x\right)}}}} - \sqrt[3]{x} \]

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-cbrt.f64 1.8

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 5.4%

   \[\leadsto \sqrt[3]{x} \]
10. 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 2024248 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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