2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.7%

Time: 4.1s

Alternatives: 14

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: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x - -1}\right)}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+152)
   (/
    1.0
    (+ (pow (cbrt (- x -1.0)) 2.0) (+ (cbrt (* x x)) (cbrt (* (- x -1.0) x)))))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+152) {
		tmp = 1.0 / (pow(cbrt((x - -1.0)), 2.0) + (cbrt((x * x)) + cbrt(((x - -1.0) * x))));
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+152) {
		tmp = 1.0 / (Math.pow(Math.cbrt((x - -1.0)), 2.0) + (Math.cbrt((x * x)) + Math.cbrt(((x - -1.0) * x))));
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+152)
		tmp = Float64(1.0 / Float64((cbrt(Float64(x - -1.0)) ^ 2.0) + Float64(cbrt(Float64(x * x)) + cbrt(Float64(Float64(x - -1.0) * x)))));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+152], N[(1.0 / N[(N[Power[N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e152

   1. Initial program 9.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 14.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 93.0%

      \[\leadsto \frac{\color{blue}{1}}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 94.4

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

   8. Applied rewrites 94.4%

      \[\leadsto \frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\color{blue}{\sqrt[3]{x \cdot x}} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. unpow1/3 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lift--.f64 99.1

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

   10. Applied rewrites 99.1%

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

   if 5e152 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. pow2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. cbrt-unprod N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 89.1

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

   6. Applied rewrites 89.1%

      \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lift-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{x \cdot x} + \frac{1}{x}\\ \frac{\mathsf{fma}\left({t\_0}^{-0.6666666666666666} \cdot \left({\left(\left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x + x\right)}\right) + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}^{-2} \cdot \frac{1}{x}\right), -0.3333333333333333, \frac{1}{\left(\sqrt[3]{t\_0} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}}\right)}{x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 2.0 (* x x)) (/ 1.0 x))))
   (/
    (fma
     (*
      (pow t_0 -0.6666666666666666)
      (*
       (pow
        (+
         (+ (pow x 0.6666666666666666) (cbrt (fma x x (+ x x))))
         (cbrt (fma x x x)))
        -2.0)
       (/ 1.0 x)))
     -0.3333333333333333
     (/
      1.0
      (+
       (+ (cbrt t_0) (cbrt (+ (/ 1.0 x) (/ 1.0 (* x x)))))
       (/ 1.0 (cbrt x)))))
    x)))

C

double code(double x) {
	double t_0 = (2.0 / (x * x)) + (1.0 / x);
	return fma((pow(t_0, -0.6666666666666666) * (pow(((pow(x, 0.6666666666666666) + cbrt(fma(x, x, (x + x)))) + cbrt(fma(x, x, x))), -2.0) * (1.0 / x))), -0.3333333333333333, (1.0 / ((cbrt(t_0) + cbrt(((1.0 / x) + (1.0 / (x * x))))) + (1.0 / cbrt(x))))) / x;
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))
	return Float64(fma(Float64((t_0 ^ -0.6666666666666666) * Float64((Float64(Float64((x ^ 0.6666666666666666) + cbrt(fma(x, x, Float64(x + x)))) + cbrt(fma(x, x, x))) ^ -2.0) * Float64(1.0 / x))), -0.3333333333333333, Float64(1.0 / Float64(Float64(cbrt(t_0) + cbrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x))))) + Float64(1.0 / cbrt(x))))) / x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Power[t$95$0, -0.6666666666666666], $MachinePrecision] * N[(N[Power[N[(N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * x + N[(x + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(1.0 / N[(N[(N[Power[t$95$0, 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.3%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 94.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666} \cdot \left({\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + {x}^{-0.3333333333333333}\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right), -0.3333333333333333, \frac{1}{\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + {x}^{-0.3333333333333333}}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. pow1/3 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-cbrt.f64 98.6

      \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666} \cdot \left({\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + {x}^{-0.3333333333333333}\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right), -0.3333333333333333, \frac{1}{\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}}\right)}{x} \]

8. Applied rewrites 98.6%

   \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666} \cdot \left({\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + {x}^{-0.3333333333333333}\right)}^{-2} \cdot \frac{1}{\left(x \cdot x\right) \cdot x}\right), -0.3333333333333333, \frac{1}{\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}}\right)}{x} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{\frac{-2}{3}} \cdot \frac{1}{x \cdot {\left(\sqrt[3]{x + {x}^{2}} + \left(\sqrt[3]{2 \cdot x + {x}^{2}} + \sqrt[3]{{x}^{2}}\right)\right)}^{2}}, \frac{-1}{3}, \frac{1}{\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}}\right)}{x} \]
10. Step-by-step derivation

    1. inv-pow N/A

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

    2. *-commutative N/A

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

    3. unpow-prod-down N/A

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

    4. inv-pow N/A

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

    5. inv-pow N/A

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

    6. lower-*.f64 N/A

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

11. Applied rewrites 98.6%

    \[\leadsto \frac{\mathsf{fma}\left({\left(\frac{2}{x \cdot x} + \frac{1}{x}\right)}^{-0.6666666666666666} \cdot \left({\left(\left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x + x\right)}\right) + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}^{-2} \cdot \frac{1}{x}\right), -0.3333333333333333, \frac{1}{\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}}\right)}{x} \]
12. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\sqrt[3]{\frac{\frac{1}{x} + 1}{x}} + {x}^{-0.3333333333333333}\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+18)
   (/
    1.0
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (* (+ (cbrt (/ (+ (/ 1.0 x) 1.0) x)) (pow x -0.3333333333333333)) x)))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 4e+18) {
		tmp = 1.0 / (pow((x - -1.0), 0.6666666666666666) + ((cbrt((((1.0 / x) + 1.0) / x)) + pow(x, -0.3333333333333333)) * x));
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+18) {
		tmp = 1.0 / (Math.pow((x - -1.0), 0.6666666666666666) + ((Math.cbrt((((1.0 / x) + 1.0) / x)) + Math.pow(x, -0.3333333333333333)) * x));
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+18)
		tmp = Float64(1.0 / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64(Float64(cbrt(Float64(Float64(Float64(1.0 / x) + 1.0) / x)) + (x ^ -0.3333333333333333)) * x)));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+18], N[(1.0 / N[(N[Power[N[(x - -1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[(N[Power[N[(N[(N[(1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, -0.3333333333333333], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e18

   1. Initial program 49.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 84.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.0%

      \[\leadsto \frac{\color{blue}{1}}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 98.0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 98.5%

       \[\leadsto \frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{x} + 1}{x}} + {x}^{-0.3333333333333333}\right) \cdot x}} \]

   if 4e18 < x

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 90.2

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

   4. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. pow2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. cbrt-unprod N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 90.2

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

   6. Applied rewrites 90.2%

      \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lift-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 4: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+14)
   (/
    1.0
    (+
     (exp (* (log (- x -1.0)) 0.6666666666666666))
     (+ (cbrt (* x x)) (cbrt (* (- x -1.0) x)))))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = 1.0 / (exp((log((x - -1.0)) * 0.6666666666666666)) + (cbrt((x * x)) + cbrt(((x - -1.0) * x))));
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = 1.0 / (Math.exp((Math.log((x - -1.0)) * 0.6666666666666666)) + (Math.cbrt((x * x)) + Math.cbrt(((x - -1.0) * x))));
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(1.0 / Float64(exp(Float64(log(Float64(x - -1.0)) * 0.6666666666666666)) + Float64(cbrt(Float64(x * x)) + cbrt(Float64(Float64(x - -1.0) * x)))));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+14], N[(1.0 / N[(N[Exp[N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 60.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 97.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.3%

      \[\leadsto \frac{\color{blue}{1}}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 98.3

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

   8. Applied rewrites 98.3%

      \[\leadsto \frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\color{blue}{\sqrt[3]{x \cdot x}} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lift--.f64 98.1

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

   10. Applied rewrites 98.1%

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

   if 5e14 < x

   1. Initial program 4.3%

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

   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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. pow2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. cbrt-unprod N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 90.3

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

   6. Applied rewrites 90.3%

      \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lift-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 5: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 2e-11)
   (* (pow (cbrt x) -2.0) 0.3333333333333333)
   (/
    1.0
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (+ (cbrt (* x x)) (cbrt (fma x x x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 2e-11) {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (pow((x - -1.0), 0.6666666666666666) + (cbrt((x * x)) + cbrt(fma(x, x, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 2e-11)
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64(cbrt(Float64(x * x)) + cbrt(fma(x, x, 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-11], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / N[(N[Power[N[(x - -1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 1.99999999999999988e-11

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. pow2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. cbrt-unprod N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 90.3

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

   6. Applied rewrites 90.3%

      \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lift-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

   if 1.99999999999999988e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 96.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.2%

      \[\leadsto \frac{\color{blue}{1}}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 98.3

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in x around 0

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

       1. lower-cbrt.f64 N/A

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

       2. +-commutative N/A

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

       3. pow2 N/A

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

       4. lower-fma.f64 98.3

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

   11. Applied rewrites 98.3%

       \[\leadsto \frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left(\sqrt[3]{x \cdot x} + \color{blue}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 2e-11)
   (* (pow (cbrt x) -2.0) 0.3333333333333333)
   (/
    1.0
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (fma x x x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 2e-11) {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	} else {
		tmp = 1.0 / (pow((x - -1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt(fma(x, x, x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 2e-11)
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	else
		tmp = Float64(1.0 / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(fma(x, x, 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-11], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(1.0 / 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]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 1.99999999999999988e-11

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. inv-pow N/A

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

      8. pow2 N/A

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

      9. unpow-prod-down N/A

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

      10. inv-pow N/A

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

      11. inv-pow N/A

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

      12. cbrt-unprod N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 N/A

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

      15. pow1/3 N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. lower-pow.f64 90.3

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

   6. Applied rewrites 90.3%

      \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. inv-pow N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lift-/.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

   if 1.99999999999999988e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 58.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 96.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in x around 0

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

      1. lower-cbrt.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 97.2

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

   9. Applied rewrites 97.2%

      \[\leadsto \frac{1}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \color{blue}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   (+
    (+
     (cbrt (+ (/ 2.0 (* x x)) (/ 1.0 x)))
     (cbrt (+ (/ 1.0 x) (/ 1.0 (* x x)))))
    (/ 1.0 (cbrt x)))
   x)))

C

double code(double x) {
	return 1.0 / (((cbrt(((2.0 / (x * x)) + (1.0 / x))) + cbrt(((1.0 / x) + (1.0 / (x * x))))) + (1.0 / cbrt(x))) * x);
}

Java

public static double code(double x) {
	return 1.0 / (((Math.cbrt(((2.0 / (x * x)) + (1.0 / x))) + Math.cbrt(((1.0 / x) + (1.0 / (x * x))))) + (1.0 / Math.cbrt(x))) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(cbrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))) + cbrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x))))) + Float64(1.0 / cbrt(x))) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[Power[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.3%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 93.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. pow1/3 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-cbrt.f64 98.1

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

8. Applied rewrites 98.1%

   \[\leadsto \frac{1}{\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \frac{1}{\sqrt[3]{x}}\right) \cdot x} \]
9. Add Preprocessing

Alternative 8: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \sqrt[3]{\frac{1}{x}}\right) \cdot x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   (+
    (+
     (cbrt (+ (/ 2.0 (* x x)) (/ 1.0 x)))
     (cbrt (+ (/ 1.0 x) (/ 1.0 (* x x)))))
    (cbrt (/ 1.0 x)))
   x)))

C

double code(double x) {
	return 1.0 / (((cbrt(((2.0 / (x * x)) + (1.0 / x))) + cbrt(((1.0 / x) + (1.0 / (x * x))))) + cbrt((1.0 / x))) * x);
}

Java

public static double code(double x) {
	return 1.0 / (((Math.cbrt(((2.0 / (x * x)) + (1.0 / x))) + Math.cbrt(((1.0 / x) + (1.0 / (x * x))))) + Math.cbrt((1.0 / x))) * x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(cbrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))) + cbrt(Float64(Float64(1.0 / x) + Float64(1.0 / Float64(x * x))))) + cbrt(Float64(1.0 / x))) * x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[Power[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

3. Applied rewrites 9.3%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 93.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-/.f64 97.9

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

8. Applied rewrites 97.9%

   \[\leadsto \frac{1}{\left(\left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{1}{x \cdot x}}\right) + \sqrt[3]{\frac{1}{x}}\right) \cdot x} \]
9. Add Preprocessing

Alternative 9: 96.3% 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 7.1%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. inv-pow N/A

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

   8. pow2 N/A

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

   9. unpow-prod-down N/A

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

   10. inv-pow N/A

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

   11. inv-pow N/A

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

   12. cbrt-unprod N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. pow1/3 N/A

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

   16. inv-pow N/A

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

   17. pow-pow N/A

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

   18. metadata-eval N/A

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

   19. lower-pow.f64 88.7

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

6. Applied rewrites 88.7%

   \[\leadsto {\left({x}^{-0.3333333333333333}\right)}^{2} \cdot 0.3333333333333333 \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. lift-/.f64 96.0

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

8. Applied rewrites 96.0%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-cbrt.f64 96.3

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

10. Applied rewrites 96.3%

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

Alternative 10: 92.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (exp (* (log 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 = exp((log(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.exp((Math.log(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(exp(Float64(log(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[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.5%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.3

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

   4. Applied rewrites 88.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 94.6

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

   6. Applied rewrites 94.6%

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

   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. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-log.f64 89.4

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

   6. Applied rewrites 89.4%

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

Alternative 11: 89.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-log.f64 89.0

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

6. Applied rewrites 89.0%

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

Alternative 12: 88.7% 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 7.1%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.7

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

4. Applied rewrites 88.7%

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

Alternative 13: 1.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around 0

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

   1. negate-sub2 N/A

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

   2. lower-neg.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-to-exp N/A

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

   5. lower-expm1.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 1.8

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

4. Applied rewrites 1.8%

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

Alternative 14: 1.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around 0

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

   1. negate-sub2 N/A

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

   2. lower-neg.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-to-exp N/A

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

   5. lower-expm1.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-log.f64 1.8

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

4. Applied rewrites 1.8%

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

5. Taylor expanded in x around inf

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

   1. lift-cbrt.f64 1.8

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

7. Applied rewrites 1.8%

   \[\leadsto -\sqrt[3]{x} \]
8. Add Preprocessing

Developer Target 1: 98.5% 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 2025116 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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