2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 94.8%

Time: 2.6s

Alternatives: 9

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% 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: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{\left(x - -1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{\left(x - -1\right) \cdot x}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+14)
   (/
    (- (- x -1.0) x)
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (* (- x -1.0) x)))))
   (if (<= x 1.32e+154)
     (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
     (* (* (pow x -0.3333333333333333) (/ 1.0 (cbrt x))) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = ((x - -1.0) - x) / (pow((x - -1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt(((x - -1.0) * x))));
	} else if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.3333333333333333) * (1.0 / cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = ((x - -1.0) - x) / (Math.pow((x - -1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) + Math.cbrt(((x - -1.0) * x))));
	} else if (x <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.3333333333333333) * (1.0 / Math.cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(Float64(Float64(x - -1.0) * x)))));
	elseif (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.3333333333333333) * Float64(1.0 / cbrt(x))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x - -1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.3333333333333333], $MachinePrecision] * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5e14

   1. Initial program 61.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 97.4%

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

   if 5e14 < x < 1.31999999999999998e154

   1. Initial program 3.7%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 1.7

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

   3. Applied rewrites 1.7%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow2 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.2

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

   6. Applied rewrites 89.2%

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

   7. Taylor expanded in x around 0

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-cbrt.f64 90.7

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

   9. Applied rewrites 90.7%

      \[\leadsto \left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 + \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (+
    (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
    (/ (* -0.1111111111111111 (cbrt x)) (* x x)))
   (* (* (pow x -0.3333333333333333) (/ 1.0 (cbrt x))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (cbrt((1.0 / (x * x))) * 0.3333333333333333) + ((-0.1111111111111111 * cbrt(x)) / (x * x));
	} else {
		tmp = (pow(x, -0.3333333333333333) * (1.0 / cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (Math.cbrt((1.0 / (x * x))) * 0.3333333333333333) + ((-0.1111111111111111 * Math.cbrt(x)) / (x * x));
	} else {
		tmp = (Math.pow(x, -0.3333333333333333) * (1.0 / Math.cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333) + Float64(Float64(-0.1111111111111111 * cbrt(x)) / Float64(x * x)));
	else
		tmp = Float64(Float64((x ^ -0.3333333333333333) * Float64(1.0 / cbrt(x))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.3333333333333333], $MachinePrecision] * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow1/3 N/A

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

      6. pow-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 88.9

          \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   4. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. div-add N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 88.9

          \[\leadsto \frac{{x}^{1.3333333333333333} \cdot 0.3333333333333333}{x \cdot x} + \frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot \color{blue}{x}} \]

   6. Applied rewrites 88.9%

      \[\leadsto \frac{{x}^{1.3333333333333333} \cdot 0.3333333333333333}{x \cdot x} + \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x}}{x \cdot x}} \]

   7. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-*.f64 96.9

         \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 + \frac{-0.1111111111111111 \cdot \color{blue}{\sqrt[3]{x}}}{x \cdot x} \]

   9. Applied rewrites 96.9%

      \[\leadsto \sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333 + \frac{\color{blue}{-0.1111111111111111 \cdot \sqrt[3]{x}}}{x \cdot x} \]

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.2

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

   6. Applied rewrites 89.2%

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

   7. Taylor expanded in x around 0

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-cbrt.f64 90.7

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

   9. Applied rewrites 90.7%

      \[\leadsto \left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (* (pow x -0.3333333333333333) (/ 1.0 (cbrt x))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.3333333333333333) * (1.0 / cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.3333333333333333) * (1.0 / Math.cbrt(x))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.3333333333333333) * Float64(1.0 / cbrt(x))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.3333333333333333], $MachinePrecision] * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow2 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.2

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

   6. Applied rewrites 89.2%

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

   7. Taylor expanded in x around 0

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lift-cbrt.f64 90.7

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

   9. Applied rewrites 90.7%

      \[\leadsto \left({x}^{-0.3333333333333333} \cdot \frac{1}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{\log x}\right)}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (pow (exp (log x)) -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = pow(exp(log(x)), -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = Math.pow(Math.exp(Math.log(x)), -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64((exp(log(x)) ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[Power[N[Exp[N[Log[x], $MachinePrecision]], $MachinePrecision], -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 9.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow2 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. 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. lower-log.f64 89.5

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

   6. Applied rewrites 89.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. exp-prod N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lift-log.f64 89.8

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

   8. Applied rewrites 89.8%

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

Alternative 5: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = (1.0 / cbrt((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 <= 1.32e+154) {
		tmp = (1.0 / Math.cbrt((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 <= 1.32e+154)
		tmp = Float64(Float64(1.0 / cbrt(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, 1.32e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $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 < 1.31999999999999998e154

   1. Initial program 9.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow2 N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 95.1

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

   8. Applied rewrites 95.1%

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

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. 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. lower-log.f64 89.5

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

   6. Applied rewrites 89.5%

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

Alternative 6: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+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 <= 1.32e+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 <= 1.32e+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, 1.32e+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 < 1.31999999999999998e154

   1. Initial program 9.0%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow1/3 N/A

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

      4. sqr-pow N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval 7.0

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

   3. Applied rewrites 7.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.9

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

   6. Applied rewrites 94.9%

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

   if 1.31999999999999998e154 < 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 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. 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. lower-log.f64 89.5

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

   6. Applied rewrites 89.5%

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

Alternative 7: 89.2% 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 6.9%

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

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 88.9

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

4. Applied rewrites 88.9%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
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. lower-log.f64 89.2

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

6. Applied rewrites 89.2%

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

Alternative 8: 88.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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 6.9%

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

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 88.9

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

4. Applied rewrites 88.9%

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

Alternative 9: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025107 
(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)))