2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 97.6%

Time: 3.6s

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333}{x \cdot x} - \frac{0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x}} \cdot \frac{0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+75)
   (-
    (/ (* (cbrt (pow x 4.0)) 0.3333333333333333) (* x x))
    (/ (* 0.1111111111111111 (cbrt x)) (* x x)))
   (* (/ 1.0 (cbrt x)) (/ 0.3333333333333333 (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+75) {
		tmp = ((cbrt(pow(x, 4.0)) * 0.3333333333333333) / (x * x)) - ((0.1111111111111111 * cbrt(x)) / (x * x));
	} else {
		tmp = (1.0 / cbrt(x)) * (0.3333333333333333 / cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+75) {
		tmp = ((Math.cbrt(Math.pow(x, 4.0)) * 0.3333333333333333) / (x * x)) - ((0.1111111111111111 * Math.cbrt(x)) / (x * x));
	} else {
		tmp = (1.0 / Math.cbrt(x)) * (0.3333333333333333 / Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+75)
		tmp = Float64(Float64(Float64(cbrt((x ^ 4.0)) * 0.3333333333333333) / Float64(x * x)) - Float64(Float64(0.1111111111111111 * cbrt(x)) / Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / cbrt(x)) * Float64(0.3333333333333333 / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+75], N[(N[(N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999985e75

   1. Initial program 14.9%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. lower--.f64 N/A

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

      15. pow-flip N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 14.9

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

   3. Applied rewrites 14.9%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

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

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

      9. *-commutative N/A

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

      10. pow2 N/A

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

      11. div-sub N/A

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

      12. lower--.f64 N/A

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

   8. Applied rewrites 95.3%

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333}{x \cdot x} - \color{blue}{\frac{0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}} \]

   if 1.99999999999999985e75 < x

   1. Initial program 4.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 37.9

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 37.9

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

   6. Applied rewrites 37.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-sqr N/A

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

      6. pow-pow N/A

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

      7. inv-pow N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. cbrt-prod N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lift-cbrt.f64 98.4

          \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

   8. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-*l/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. pow1/3 N/A

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

      9. pow-neg N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-neg N/A

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

      16. pow1/3 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cbrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (pow x -5.0))
  -0.1111111111111111
  (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(cbrt(pow(x, -5.0)), -0.1111111111111111, (pow(cbrt(x), -2.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64((cbrt(x) ^ -2.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. rgt-mult-inverse N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower--.f64 N/A

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

   15. pow-flip N/A

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

   16. lower-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower--.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 24.3

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

6. Applied rewrites 24.3%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-flip N/A

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

   11. metadata-eval N/A

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

   12. pow-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow1/3 N/A

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

   17. lower-pow.f64 N/A

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

   18. lift-cbrt.f64 97.5

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]

9. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]
10. Add Preprocessing

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x}} \cdot \frac{0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+75)
   (/
    (fma
     (cbrt (* (* x x) (* x x)))
     0.3333333333333333
     (* -0.1111111111111111 (cbrt x)))
    (* x x))
   (* (/ 1.0 (cbrt x)) (/ 0.3333333333333333 (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+75) {
		tmp = fma(cbrt(((x * x) * (x * x))), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))) / (x * x);
	} else {
		tmp = (1.0 / cbrt(x)) * (0.3333333333333333 / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+75)
		tmp = Float64(fma(cbrt(Float64(Float64(x * x) * Float64(x * x))), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / cbrt(x)) * Float64(0.3333333333333333 / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+75], N[(N[(N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999985e75

   1. Initial program 14.9%

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

      1. lift-+.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. rgt-mult-inverse N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. flip-+ N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-times N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. lower--.f64 N/A

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

      15. pow-flip N/A

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

      16. lower-pow.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. inv-pow N/A

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

      20. lower-pow.f64 14.9

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

   3. Applied rewrites 14.9%

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

   4. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.3

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

   6. Applied rewrites 95.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 95.3

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

   8. Applied rewrites 95.3%

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

   if 1.99999999999999985e75 < x

   1. Initial program 4.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 37.9

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 37.9%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 37.9

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

   6. Applied rewrites 37.9%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-sqr N/A

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

      6. pow-pow N/A

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

      7. inv-pow N/A

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

      8. cbrt-div N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. cbrt-prod N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. lift-cbrt.f64 98.4

          \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

   8. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. associate-*l/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. pow1/3 N/A

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

      9. pow-neg N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. pow-neg N/A

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

      16. pow1/3 N/A

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

      17. lower-/.f64 N/A

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

      18. lift-cbrt.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 4: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lower-*.f64 50.8

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

6. Applied rewrites 50.8%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-sqr N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. cbrt-prod N/A

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

   12. lower-/.f64 N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-cbrt.f64 96.5

       \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

8. Applied rewrites 96.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. associate-*l/ N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. pow1/3 N/A

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

   9. pow-neg N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-neg N/A

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

   16. pow1/3 N/A

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

   17. lower-/.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lift-cbrt.f64 96.5

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

10. Applied rewrites 96.5%

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

Alternative 5: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow (cbrt x) -2.0) 0.3333333333333333))

C

double code(double x) {
	return pow(cbrt(x), -2.0) * 0.3333333333333333;
}

Java

public static double code(double x) {
	return Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
}

Julia

function code(x)
	return Float64((cbrt(x) ^ -2.0) * 0.3333333333333333)
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. rgt-mult-inverse N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. lower--.f64 N/A

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

   15. pow-flip N/A

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

   16. lower-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower--.f64 N/A

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

   19. inv-pow N/A

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

   20. lower-pow.f64 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 24.3

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

6. Applied rewrites 24.3%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. pow1/3 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-cbrt.f64 96.5

       \[\leadsto {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333 \]

9. Applied rewrites 96.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333} \]
10. Add Preprocessing

Alternative 6: 92.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 94.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.0

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

   6. Applied rewrites 95.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 7.8

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

   6. Applied rewrites 7.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 89.1

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

   8. Applied rewrites 89.1%

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

Alternative 7: 91.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.3%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 94.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 94.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 94.8

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

   6. Applied rewrites 94.8%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   4. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. lift-pow.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lower-*.f64 7.8

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

   6. Applied rewrites 7.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 89.1

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

   8. Applied rewrites 89.1%

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

Alternative 8: 88.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lower-*.f64 50.8

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

6. Applied rewrites 50.8%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-sqr N/A

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

   6. pow-pow N/A

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

   7. inv-pow N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. unpow2 N/A

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

   11. cbrt-prod N/A

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

   12. lower-/.f64 N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-cbrt.f64 96.5

       \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

8. Applied rewrites 96.5%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-pow N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 88.8

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

10. Applied rewrites 88.8%

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

Alternative 9: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

4. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. lift-pow.f64 N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lower-*.f64 50.8

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

6. Applied rewrites 50.8%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 88.8

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

8. Applied rewrites 88.8%

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

Alternative 10: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025095 
(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)))