2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.3%

Time: 3.3s

Alternatives: 10

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.1% 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: 98.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+14)
   (/
    (+ (- x -1.0) (- x))
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (- (pow x 0.6666666666666666) (* (cbrt (- x -1.0)) (- (cbrt x))))))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+14) {
		tmp = ((x - -1.0) + -x) / (pow((x - -1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) - (cbrt((x - -1.0)) * -cbrt(x))));
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e+14) {
		tmp = ((x - -1.0) + -x) / (Math.pow((x - -1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) - (Math.cbrt((x - -1.0)) * -Math.cbrt(x))));
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+14)
		tmp = Float64(Float64(Float64(x - -1.0) + Float64(-x)) / Float64((Float64(x - -1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) - Float64(cbrt(Float64(x - -1.0)) * Float64(-cbrt(x))))));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+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[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] * (-N[Power[x, 1/3], $MachinePrecision])), $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 < 1e14

   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. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. flip3-+ N/A

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

      8. lower-/.f64 N/A

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

      9. rem-cube-cbrt N/A

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

      10. mul-1-neg N/A

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

      11. cube-neg N/A

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

      12. rem-cube-cbrt N/A

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

      13. lower-+.f64 N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 N/A

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

   3. Applied rewrites 9.3%

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

   if 1e14 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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

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

      3. pow-flip N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow1/3 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. metadata-eval 96.4

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

   6. Applied rewrites 96.4%

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

Alternative 2: 98.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+14)
   (/
    (- (- x -1.0) x)
    (+
     (pow (- x -1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (* (- x -1.0) x)))))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+14) {
		tmp = ((x - -1.0) - x) / (pow((x - -1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + 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 <= 1e+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 {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+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)))));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+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], 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 < 1e14

   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. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. 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%

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

   if 1e14 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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

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

      3. pow-flip N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow1/3 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. metadata-eval 96.4

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

   6. Applied rewrites 96.4%

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

Alternative 3: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 38000000:\\ \;\;\;\;{\left(x - -1\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 38000000.0)
   (- (pow (- x -1.0) 0.3333333333333333) (pow x 0.3333333333333333))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 38000000.0) {
		tmp = pow((x - -1.0), 0.3333333333333333) - pow(x, 0.3333333333333333);
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 38000000.0) {
		tmp = Math.pow((x - -1.0), 0.3333333333333333) - Math.pow(x, 0.3333333333333333);
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 38000000.0)
		tmp = Float64((Float64(x - -1.0) ^ 0.3333333333333333) - (x ^ 0.3333333333333333));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 38000000.0], N[(N[Power[N[(x - -1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 0.3333333333333333], $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 < 3.8e7

   1. Initial program 7.1%

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

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

      5. add-flip N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 4.7

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

   3. Applied rewrites 4.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 7.2

         \[\leadsto {\left(x - -1\right)}^{0.3333333333333333} - \color{blue}{{x}^{0.3333333333333333}} \]

   5. Applied rewrites 7.2%

      \[\leadsto {\left(x - -1\right)}^{0.3333333333333333} - \color{blue}{{x}^{0.3333333333333333}} \]

   if 3.8e7 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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

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

      3. pow-flip N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow1/3 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. metadata-eval 96.4

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

   6. Applied rewrites 96.4%

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

Alternative 4: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 30000000:\\ \;\;\;\;\sqrt[3]{x - -1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 30000000.0)
   (- (cbrt (- x -1.0)) (cbrt x))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 30000000.0) {
		tmp = cbrt((x - -1.0)) - cbrt(x);
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 30000000.0) {
		tmp = Math.cbrt((x - -1.0)) - Math.cbrt(x);
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 30000000.0)
		tmp = Float64(cbrt(Float64(x - -1.0)) - cbrt(x));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 30000000.0], N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $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 < 3e7

   1. Initial program 7.1%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 7.1

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

   3. Applied rewrites 7.1%

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

   if 3e7 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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

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

      3. pow-flip N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. pow-prod-up N/A

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

      7. pow1/3 N/A

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

      8. pow1/3 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. metadata-eval 96.4

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

   6. Applied rewrites 96.4%

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

Alternative 5: 96.4% 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

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. pow1/3 N/A

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

   4. pow1/3 N/A

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

   5. mult-flip-rev N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. 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

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

   3. pow-flip N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow1/3 N/A

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

   8. pow1/3 N/A

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

   9. unpow-prod-down N/A

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

   10. pow-prod-up N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-cbrt.f64 N/A

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

   15. metadata-eval 96.4

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

6. Applied rewrites 96.4%

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

Alternative 6: 92.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{e^{\frac{\log x \cdot 2}{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (/ 0.3333333333333333 (exp (/ (* (log x) 2.0) 3.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / exp(((log(x) * 2.0) / 3.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / Math.exp(((Math.log(x) * 2.0) / 3.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 / exp(Float64(Float64(log(x) * 2.0) / 3.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Exp[N[(N[(N[Log[x], $MachinePrecision] * 2.0), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

      \[\leadsto \frac{0.3333333333333333}{\sqrt[3]{x \cdot x}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow2 N/A

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

      4. pow-to-exp N/A

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

      5. exp-cbrt-rev N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 89.5

         \[\leadsto \frac{0.3333333333333333}{e^{\frac{\log x \cdot 2}{3}}} \]

   10. Applied rewrites 89.5%

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

Alternative 7: 92.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} 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 = 0.3333333333333333 / Math.cbrt((x * x));
	} 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(0.3333333333333333 / cbrt(Float64(x * x)));
	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[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $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 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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. lower-log.f64 89.0

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

   6. Applied rewrites 89.0%

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

Alternative 8: 92.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. 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-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. mult-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f64 88.7

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow1/3 N/A

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

      4. pow1/3 N/A

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

      5. mult-flip-rev N/A

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 88.7

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

   4. Applied rewrites 88.7%

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

Alternative 9: 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

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. pow1/3 N/A

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

   4. pow1/3 N/A

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

   5. mult-flip-rev N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. 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 10: 4.1% accurate, 36.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

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 = 0.0d0
end function

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{0} \]
3. Step-by-step derivation

4. Applied rewrites 4.1%

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

Developer Target 1: 98.4% 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 2025140 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  :herbie-expected 5/2

  :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)))