2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 98.4%

Time: 2.8s

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{t\_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (if (<= x 7.2e+15)
     (/ (- (- x -1.0) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0))))
     (* (* (/ 1.0 (cbrt x)) (cbrt (pow x -1.0))) 0.3333333333333333))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	double tmp;
	if (x <= 7.2e+15) {
		tmp = ((x - -1.0) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	} else {
		tmp = ((1.0 / cbrt(x)) * cbrt(pow(x, -1.0))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x - -1.0));
	double tmp;
	if (x <= 7.2e+15) {
		tmp = ((x - -1.0) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	} else {
		tmp = ((1.0 / Math.cbrt(x)) * Math.cbrt(Math.pow(x, -1.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 7.2e+15)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	else
		tmp = Float64(Float64(Float64(1.0 / cbrt(x)) * cbrt((x ^ -1.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 7.2e+15], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e15

   1. Initial program 57.1%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   3. Applied rewrites 98.7%

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

   if 7.2e15 < x

   1. Initial program 4.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 50.3

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

   6. Applied rewrites 50.3%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. inv-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. inv-pow N/A

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

      6. lower-*.f64 N/A

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

      7. cbrt-prod N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. inv-pow N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-cbrt.f64 N/A

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

      13. inv-pow N/A

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

      14. lift-pow.f64 98.0

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

   8. Applied rewrites 98.0%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. inv-pow N/A

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

      4. cbrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-cbrt.f64 98.4

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

   10. Applied rewrites 98.4%

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

Alternative 2: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333 \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 50.9

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

6. Applied rewrites 50.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. inv-pow N/A

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

   4. lift-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-*.f64 N/A

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

   7. cbrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. inv-pow N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. inv-pow N/A

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

   14. lift-pow.f64 96.3

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

8. Applied rewrites 96.3%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. inv-pow N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lift-cbrt.f64 96.7

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

10. Applied rewrites 96.7%

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

Alternative 3: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt[3]{x}}\\ t\_0 \cdot \left(t\_0 \cdot 0.3333333333333333\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (cbrt x)))) (* t_0 (* t_0 0.3333333333333333))))

C

double code(double x) {
	double t_0 = 1.0 / cbrt(x);
	return t_0 * (t_0 * 0.3333333333333333);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.cbrt(x);
	return t_0 * (t_0 * 0.3333333333333333);
}

Julia

function code(x)
	t_0 = Float64(1.0 / cbrt(x))
	return Float64(t_0 * Float64(t_0 * 0.3333333333333333))
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 50.9

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

6. Applied rewrites 50.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. inv-pow N/A

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

   4. lift-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-*.f64 N/A

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

   7. cbrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. inv-pow N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. inv-pow N/A

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

   14. lift-pow.f64 96.3

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

8. Applied rewrites 96.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. inv-pow N/A

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

   8. cbrt-div N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-*.f64 96.7

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

   13. lift-cbrt.f64 N/A

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

   14. lift-pow.f64 N/A

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

   15. inv-pow N/A

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

   16. cbrt-div N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. lift-cbrt.f64 96.6

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

10. Applied rewrites 96.6%

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

Alternative 4: 96.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (/ 1.0 (cbrt x)) 2.0) 0.3333333333333333))

C

double code(double x) {
	return pow((1.0 / cbrt(x)), 2.0) * 0.3333333333333333;
}

Java

public static double code(double x) {
	return Math.pow((1.0 / Math.cbrt(x)), 2.0) * 0.3333333333333333;
}

Julia

function code(x)
	return Float64((Float64(1.0 / cbrt(x)) ^ 2.0) * 0.3333333333333333)
end

Wolfram

code[x_] := N[(N[Power[N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 50.9

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

6. Applied rewrites 50.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. inv-pow N/A

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

   4. lift-pow.f64 N/A

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

   5. inv-pow N/A

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

   6. lower-*.f64 N/A

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

   7. cbrt-prod N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cbrt.f64 N/A

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

   10. inv-pow N/A

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

   11. lift-pow.f64 N/A

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

   12. lower-cbrt.f64 N/A

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

   13. inv-pow N/A

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

   14. lift-pow.f64 96.3

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

8. Applied rewrites 96.3%

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

   1. lift-*.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. inv-pow N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. inv-pow N/A

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

   8. cbrt-prod N/A

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

   9. inv-pow N/A

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

   10. lift-pow.f64 N/A

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

   11. inv-pow N/A

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

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. lift-pow.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. lift-cbrt.f64 50.9

       \[\leadsto \sqrt[3]{{x}^{-1} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)} \cdot 0.3333333333333333 \]

10. Applied rewrites 96.6%

    \[\leadsto \color{blue}{{\left(\frac{1}{\sqrt[3]{x}}\right)}^{2} \cdot 0.3333333333333333} \]
11. Add Preprocessing

Alternative 5: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[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 8.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.5

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

   4. Applied rewrites 7.5%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 7.5

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

   6. Applied rewrites 7.5%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 89.2

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

   8. Applied rewrites 89.2%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-log.f64 89.5

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

   10. Applied rewrites 89.5%

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

Alternative 6: 92.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.4

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

   6. Applied rewrites 95.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.5

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

   4. Applied rewrites 7.5%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 7.5

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

   6. Applied rewrites 7.5%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 89.2

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

   8. Applied rewrites 89.2%

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

Alternative 7: 92.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 95.2

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

   6. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.5

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

   4. Applied rewrites 7.5%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

      3. metadata-eval N/A

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

      4. inv-pow N/A

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

      5. metadata-eval N/A

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

      6. inv-pow N/A

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

      7. lower-*.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 7.5

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

   6. Applied rewrites 7.5%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 89.2

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

   8. Applied rewrites 89.2%

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

Alternative 8: 89.0% 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 6.7%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 50.9

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

4. Applied rewrites 50.9%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. metadata-eval N/A

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

   6. inv-pow N/A

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

   7. lower-*.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 50.9

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

6. Applied rewrites 50.9%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 89.0

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

8. Applied rewrites 89.0%

   \[\leadsto {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]
9. 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 6.7%

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

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 2025093 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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