2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 98.9%

Time: 11.0s

Alternatives: 10

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 4.5e14

   1. Initial program 62.7%

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

   4. Applied rewrites 97.5%

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

   if 4.5e14 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-cbrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. lift-cbrt.f64 N/A

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

      16. lower-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/3 N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow2 N/A

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

      11. lift-/.f64 N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. pow-plus N/A

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

      21. pow-pow N/A

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

      22. pow-flip N/A

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

      23. lift-pow.f64 N/A

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

      24. lift-/.f64 N/A

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

      25. pow1/3 N/A

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

      26. lift-cbrt.f64 N/A

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

      27. *-commutative N/A

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

   9. Applied rewrites 50.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{x}^{1.6666666666666667}}{x}}} \]

   11. Applied rewrites 99.1%

       \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{x}{\sqrt[3]{x} + 0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

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

Alternative 2: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + -1\right)}^{-0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 2e-7)
   (* 0.3333333333333333 (/ 1.0 (/ x (cbrt x))))
   (-
    (* (pow (+ x -1.0) -0.3333333333333333) (cbrt (fma x x -1.0)))
    (cbrt x))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 2e-7) {
		tmp = 0.3333333333333333 * (1.0 / (x / cbrt(x)));
	} else {
		tmp = (pow((x + -1.0), -0.3333333333333333) * cbrt(fma(x, x, -1.0))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 2e-7)
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(x / cbrt(x))));
	else
		tmp = Float64(Float64((Float64(x + -1.0) ^ -0.3333333333333333) * cbrt(fma(x, 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], 2e-7], N[(0.3333333333333333 * N[(1.0 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x + -1.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision] * N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $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)) < 1.9999999999999999e-7

   1. Initial program 5.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-cbrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. lift-cbrt.f64 N/A

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

      16. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/3 N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow2 N/A

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

      11. lift-/.f64 N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. pow-plus N/A

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

      21. pow-pow N/A

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

      22. pow-flip N/A

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

      23. lift-pow.f64 N/A

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

      24. lift-/.f64 N/A

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

      25. pow1/3 N/A

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

      26. lift-cbrt.f64 N/A

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

      27. *-commutative N/A

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

   9. Applied rewrites 51.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{x}^{1.6666666666666667}}{x}}} \]

   11. Applied rewrites 98.4%

       \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{x}{\sqrt[3]{x} + 0}}} \]

   if 1.9999999999999999e-7 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 87.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. cbrt-prod N/A

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

      5. lower-*.f64 N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. sub-neg N/A

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

      12. lower-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-cbrt.f64 N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 89.5

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

   4. Applied rewrites 89.5%

      \[\leadsto \color{blue}{{\left(x + -1\right)}^{-0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x + -1\right)}^{-0.3333333333333333} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.16666666666666666}, -{x}^{0.16666666666666666}, \frac{1}{{\left(x + 1\right)}^{-0.3333333333333333}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (fma
    (pow x 0.16666666666666666)
    (- (pow x 0.16666666666666666))
    (/ 1.0 (pow (+ x 1.0) -0.3333333333333333)))
   (* 0.3333333333333333 (/ 1.0 (/ x (cbrt x))))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = fma(pow(x, 0.16666666666666666), -pow(x, 0.16666666666666666), (1.0 / pow((x + 1.0), -0.3333333333333333)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / (x / cbrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = fma((x ^ 0.16666666666666666), Float64(-(x ^ 0.16666666666666666)), Float64(1.0 / (Float64(x + 1.0) ^ -0.3333333333333333)));
	else
		tmp = Float64(0.3333333333333333 * Float64(1.0 / Float64(x / cbrt(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[x, 0.16666666666666666], $MachinePrecision] * (-N[Power[x, 0.16666666666666666], $MachinePrecision]) + N[(1.0 / N[Power[N[(x + 1.0), $MachinePrecision], -0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(1.0 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 87.1%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. sqr-pow N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 87.3

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

   4. Applied rewrites 87.3%

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

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \sqrt[3]{\color{blue}{x + 1}}\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \sqrt[3]{\color{blue}{\frac{1}{\frac{1}{x + 1}}}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \sqrt[3]{\frac{1}{\color{blue}{\frac{1}{x + 1}}}}\right) \]

      4. cbrt-div N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{1}{x + 1}}}}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{\color{blue}{1}}{\sqrt[3]{\frac{1}{x + 1}}}\right) \]

      6. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{\color{blue}{\sqrt[3]{\frac{1}{x + 1}}}}\right) \]

      7. lift-/.f64 86.2

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

      8. lift-cbrt.f64 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{\color{blue}{\sqrt[3]{\frac{1}{x + 1}}}}\right) \]

      9. pow1/3 N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{\color{blue}{{\left(\frac{1}{x + 1}\right)}^{\frac{1}{3}}}}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{{\color{blue}{\left(\frac{1}{x + 1}\right)}}^{\frac{1}{3}}}\right) \]

      11. inv-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{{\color{blue}{\left({\left(x + 1\right)}^{-1}\right)}}^{\frac{1}{3}}}\right) \]

      12. pow-pow N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{\color{blue}{{\left(x + 1\right)}^{\left(-1 \cdot \frac{1}{3}\right)}}}\right) \]

      13. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{\color{blue}{{\left(x + 1\right)}^{\left(-1 \cdot \frac{1}{3}\right)}}}\right) \]

      14. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{{\color{blue}{\left(x + 1\right)}}^{\left(-1 \cdot \frac{1}{3}\right)}}\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{{\color{blue}{\left(1 + x\right)}}^{\left(-1 \cdot \frac{1}{3}\right)}}\right) \]

      16. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \mathsf{neg}\left({x}^{\frac{1}{6}}\right), \frac{1}{{\color{blue}{\left(1 + x\right)}}^{\left(-1 \cdot \frac{1}{3}\right)}}\right) \]

      17. metadata-eval 90.4

          \[\leadsto \mathsf{fma}\left({x}^{0.16666666666666666}, -{x}^{0.16666666666666666}, \frac{1}{{\left(1 + x\right)}^{\color{blue}{-0.3333333333333333}}}\right) \]

   6. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left({x}^{0.16666666666666666}, -{x}^{0.16666666666666666}, \color{blue}{\frac{1}{{\left(1 + x\right)}^{-0.3333333333333333}}}\right) \]

   if 3.3e7 < x

   1. Initial program 5.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 46.8

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

   5. Applied rewrites 46.8%

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

      1. metadata-eval N/A

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

      2. frac-times N/A

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

      3. cbrt-prod N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. lower-*.f64 N/A

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

      7. pow1/3 N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. lift-cbrt.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. cbrt-div N/A

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

      14. metadata-eval N/A

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

      15. lift-cbrt.f64 N/A

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

      16. lower-/.f64 97.8

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

   7. Applied rewrites 97.8%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

      3. lift-/.f64 N/A

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

      4. pow1/3 N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. lift-/.f64 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow2 N/A

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

      11. lift-/.f64 N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. pow-pow N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. pow-plus N/A

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

      21. pow-pow N/A

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

      22. pow-flip N/A

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

      23. lift-pow.f64 N/A

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

      24. lift-/.f64 N/A

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

      25. pow1/3 N/A

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

      26. lift-cbrt.f64 N/A

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

      27. *-commutative N/A

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

   9. Applied rewrites 51.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{x}^{1.6666666666666667}}{x}}} \]

   11. Applied rewrites 98.4%

       \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{x}{\sqrt[3]{x} + 0}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.16666666666666666}, -{x}^{0.16666666666666666}, \frac{1}{{\left(x + 1\right)}^{-0.3333333333333333}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / (x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / 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((1.0 / (x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / 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(1.0 / Float64(x * x))));
	else
		tmp = Float64(0.3333333333333333 * Float64(1.0 / (x ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 11.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

   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. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. cbrt-div N/A

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

      3. metadata-eval N/A

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

      4. lift-*.f64 N/A

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

      5. cbrt-prod N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. pow1/3 N/A

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

      11. lift-cbrt.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 89.1

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

   7. Applied rewrites 89.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.6666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 46.4

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

5. Applied rewrites 46.4%

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

   1. metadata-eval N/A

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

   2. frac-times N/A

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

   3. cbrt-prod N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. lower-*.f64 N/A

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

   7. pow1/3 N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. lift-cbrt.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow1/3 N/A

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

   13. cbrt-div N/A

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

   14. metadata-eval N/A

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

   15. lift-cbrt.f64 N/A

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

   16. lower-/.f64 96.0

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

7. Applied rewrites 96.0%

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

   1. metadata-eval N/A

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

   2. cbrt-div N/A

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

   3. lift-/.f64 N/A

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

   4. pow1/3 N/A

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

   5. metadata-eval N/A

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

   6. cbrt-div N/A

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

   7. lift-/.f64 N/A

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

   8. pow1/3 N/A

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

   9. unpow-prod-down N/A

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

   10. pow2 N/A

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

   11. lift-/.f64 N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. pow-pow N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. pow-plus N/A

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

   21. pow-pow N/A

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

   22. pow-flip N/A

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

   23. lift-pow.f64 N/A

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

   24. lift-/.f64 N/A

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

   25. pow1/3 N/A

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

   26. lift-cbrt.f64 N/A

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

   27. *-commutative N/A

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

9. Applied rewrites 50.7%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{{x}^{1.6666666666666667}}{x}}} \]

11. Applied rewrites 96.6%

    \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{\frac{x}{\sqrt[3]{x} + 0}}} \]

12. Final simplification 96.6%

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

Alternative 6: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{1}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 46.4

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

5. Applied rewrites 46.4%

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

   1. lift-*.f64 N/A

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

   2. cbrt-div N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. cbrt-prod N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-cbrt.f64 N/A

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

   10. pow1/3 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. pow1/3 N/A

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

   13. pow-prod-up N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 88.2

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

7. Applied rewrites 88.2%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 7: 89.0% 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

real(8) function code(x)
    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 7.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 46.4

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

5. Applied rewrites 46.4%

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

   1. lift-*.f64 N/A

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

   2. cbrt-div N/A

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

   3. metadata-eval N/A

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

   4. lift-*.f64 N/A

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

   5. cbrt-prod N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. pow1/3 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. pow1/3 N/A

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

   14. pow-prod-up N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 88.2

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

7. Applied rewrites 88.2%

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

Alternative 8: 89.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    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 7.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 46.4

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

5. Applied rewrites 46.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 46.4

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-pow N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 88.2

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

7. Applied rewrites 88.2%

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

8. Final simplification 88.2%

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

Alternative 9: 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 7.8%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cbrt.f64 1.8

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

5. Applied rewrites 1.8%

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

7. Applied rewrites 5.3%

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

8. Taylor expanded in x around inf

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

   1. lower-cbrt.f64 5.3

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

10. Applied rewrites 5.3%

    \[\leadsto \color{blue}{\sqrt[3]{x}} \]
11. Add Preprocessing

Alternative 10: 4.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.8%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. sqr-pow N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval 8.6

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

4. Applied rewrites 8.6%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. mul0-lft N/A

      \[\leadsto x \cdot \color{blue}{0} \]

   4. mul0-rgt 4.2

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 4.2%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(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)))