2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 99.1%

Time: 3.8s

Alternatives: 11

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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{\left(\sqrt[3]{x - -1}\right)}^{2} + \mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\left(x - -1\right) \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+14)
   (/
    (- (- x -1.0) x)
    (+
     (pow (cbrt (- x -1.0)) 2.0)
     (fma (cbrt x) (cbrt x) (cbrt (* (- x -1.0) x)))))
   (fma
    (cbrt x)
    (/ 0.3333333333333333 x)
    (* (/ (/ (cbrt x) 1.0) x) (/ 0.1111111111111111 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = ((x - -1.0) - x) / (pow(cbrt((x - -1.0)), 2.0) + fma(cbrt(x), cbrt(x), cbrt(((x - -1.0) * x))));
	} else {
		tmp = fma(cbrt(x), (0.3333333333333333 / x), (((cbrt(x) / 1.0) / x) * (0.1111111111111111 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((cbrt(Float64(x - -1.0)) ^ 2.0) + fma(cbrt(x), cbrt(x), cbrt(Float64(Float64(x - -1.0) * x)))));
	else
		tmp = fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(Float64(cbrt(x) / 1.0) / x) * Float64(0.1111111111111111 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+14], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(N[(x - -1.0), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] / 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 61.0%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 58.2

         \[\leadsto \sqrt[3]{x + 1} - e^{\color{blue}{\log x} \cdot 0.3333333333333333} \]

   4. Applied rewrites 58.2%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{e^{\log x \cdot 0.3333333333333333}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-exp.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. exp-to-pow N/A

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

      8. pow1/3 N/A

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

      9. flip3-- N/A

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

      10. lower-/.f64 N/A

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

      11. rem-cube-cbrt N/A

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

      12. rem-cube-cbrt N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower--.f64 N/A

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

   6. Applied rewrites 99.0%

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

   if 5e14 < x

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right) \cdot \frac{1}{3}}{{x}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{{x}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{2.5}}, 0.1111111111111111, \sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{\color{blue}{x \cdot x}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{\color{blue}{x} \cdot x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      8. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{\color{blue}{x} \cdot x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{x \cdot x} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{\color{blue}{2}}} \]

      13. div-add N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}} + \color{blue}{\frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{2}}} \]

   7. Applied rewrites 23.0%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   8. Taylor expanded in x around 0

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

      1. lift-cbrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{t\_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (if (<= x 5e+14)
     (/ (- (- x -1.0) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0))))
     (fma
      (cbrt x)
      (/ 0.3333333333333333 x)
      (* (/ (/ (cbrt x) 1.0) x) (/ 0.1111111111111111 x))))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	double tmp;
	if (x <= 5e+14) {
		tmp = ((x - -1.0) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	} else {
		tmp = fma(cbrt(x), (0.3333333333333333 / x), (((cbrt(x) / 1.0) / x) * (0.1111111111111111 / x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(x - -1.0) - x) / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	else
		tmp = fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(Float64(cbrt(x) / 1.0) / x) * Float64(0.1111111111111111 / x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 5e+14], 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[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] / 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 61.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing
   3. 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

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

   4. 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 5e14 < x

   1. Initial program 4.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right) \cdot \frac{1}{3}}{{x}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{{x}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 21.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{2.5}}, 0.1111111111111111, \sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{\color{blue}{x \cdot x}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{\color{blue}{x} \cdot x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      8. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{\color{blue}{x} \cdot x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{x \cdot x} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{\color{blue}{2}}} \]

      13. div-add N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}} + \color{blue}{\frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{2}}} \]

   7. Applied rewrites 23.0%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   8. Taylor expanded in x around 0

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

      1. lift-cbrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \sqrt[3]{{x}^{-2}} \cdot 0.06172839506172839\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+73)
   (/
    (fma
     -0.1111111111111111
     (cbrt x)
     (fma
      (cbrt (pow x 4.0))
      0.3333333333333333
      (* (cbrt (pow x -2.0)) 0.06172839506172839)))
    (* x x))
   (fma
    (cbrt x)
    (/ 0.3333333333333333 x)
    (* (/ (/ (cbrt x) 1.0) x) (/ 0.1111111111111111 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+73) {
		tmp = fma(-0.1111111111111111, cbrt(x), fma(cbrt(pow(x, 4.0)), 0.3333333333333333, (cbrt(pow(x, -2.0)) * 0.06172839506172839))) / (x * x);
	} else {
		tmp = fma(cbrt(x), (0.3333333333333333 / x), (((cbrt(x) / 1.0) / x) * (0.1111111111111111 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+73)
		tmp = Float64(fma(-0.1111111111111111, cbrt(x), fma(cbrt((x ^ 4.0)), 0.3333333333333333, Float64(cbrt((x ^ -2.0)) * 0.06172839506172839))) / Float64(x * x));
	else
		tmp = fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(Float64(cbrt(x) / 1.0) / x) * Float64(0.1111111111111111 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+73], N[(N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision] + N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] / 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999983e72

   1. Initial program 15.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. 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 90.4

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   7. Step-by-step derivation

      1. pow1/3 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \color{blue}{\left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}}{{x}^{2}} \]

      2. exp-to-pow N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \color{blue}{\left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}}{{x}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   8. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \sqrt[3]{{x}^{-2}} \cdot 0.06172839506172839\right)\right)}{x \cdot x}} \]

   if 9.99999999999999983e72 < x

   1. Initial program 4.4%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right) \cdot \frac{1}{3}}{{x}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{{x}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{2.5}}, 0.1111111111111111, \sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{\color{blue}{x \cdot x}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{\color{blue}{x} \cdot x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      8. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{\color{blue}{x} \cdot x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{x \cdot x} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{\color{blue}{2}}} \]

      13. div-add N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}} + \color{blue}{\frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{2}}} \]

   7. Applied rewrites 4.0%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   8. Taylor expanded in x around 0

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

      1. lift-cbrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+73)
   (/
    (fma
     (cbrt (pow x -2.0))
     0.06172839506172839
     (fma
      (cbrt (pow x 4.0))
      0.3333333333333333
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (fma
    (cbrt x)
    (/ 0.3333333333333333 x)
    (* (/ (/ (cbrt x) 1.0) x) (/ 0.1111111111111111 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+73) {
		tmp = fma(cbrt(pow(x, -2.0)), 0.06172839506172839, fma(cbrt(pow(x, 4.0)), 0.3333333333333333, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = fma(cbrt(x), (0.3333333333333333 / x), (((cbrt(x) / 1.0) / x) * (0.1111111111111111 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+73)
		tmp = Float64(fma(cbrt((x ^ -2.0)), 0.06172839506172839, fma(cbrt((x ^ 4.0)), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(Float64(cbrt(x) / 1.0) / x) * Float64(0.1111111111111111 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+73], N[(N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] / 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999983e72

   1. Initial program 15.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   if 9.99999999999999983e72 < x

   1. Initial program 4.4%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right) \cdot \frac{1}{3}}{{x}^{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{{x}^{2}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 2.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{2.5}}, 0.1111111111111111, \sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}{x \cdot x}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{x \cdot \color{blue}{x}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{\color{blue}{x \cdot x}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{\color{blue}{x} \cdot x} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      5. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      8. lift-cbrt.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      9. lift-pow.f64 N/A

         \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

      10. +-commutative N/A

          \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{\color{blue}{x} \cdot x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{x \cdot x} \]

      12. pow2 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{\color{blue}{2}}} \]

      13. div-add N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}} + \color{blue}{\frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{2}}} \]

   7. Applied rewrites 4.0%

      \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   8. Taylor expanded in x around 0

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

      1. lift-cbrt.f64 99.1

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt x)
  (/ 0.3333333333333333 x)
  (* (/ (/ (cbrt x) 1.0) x) (/ 0.1111111111111111 x))))

C

double code(double x) {
	return fma(cbrt(x), (0.3333333333333333 / x), (((cbrt(x) / 1.0) / x) * (0.1111111111111111 / x)));
}

Julia

function code(x)
	return fma(cbrt(x), Float64(0.3333333333333333 / x), Float64(Float64(Float64(cbrt(x) / 1.0) / x) * Float64(0.1111111111111111 / x)))
end

Wolfram

code[x_] := N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision] + N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] / 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around -inf

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

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right) \cdot \frac{1}{3}}{{x}^{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{{x}^{2}} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{9} \cdot \left(\sqrt[3]{x} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{5}}\right) + \frac{1}{3} \cdot \left(\sqrt[3]{{x}^{4}} \cdot \frac{1}{{\left(\sqrt[3]{-1}\right)}^{2}}\right)}{\color{blue}{{x}^{2}}} \]

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{2.5}}, 0.1111111111111111, \sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}{x \cdot x}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{x \cdot \color{blue}{x}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}}, \frac{1}{9}, \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}\right)}{\color{blue}{x \cdot x}} \]

   3. lift-fma.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{\color{blue}{x} \cdot x} \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   5. lift-cbrt.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   6. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   8. lift-cbrt.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   9. lift-pow.f64 N/A

      \[\leadsto \frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9} + \sqrt[3]{{x}^{4}} \cdot \frac{1}{3}}{x \cdot x} \]

   10. +-commutative N/A

       \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{\color{blue}{x} \cdot x} \]

   11. *-commutative N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{x \cdot x} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{\color{blue}{2}}} \]

   13. div-add N/A

       \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}} + \color{blue}{\frac{\frac{\sqrt[3]{x}}{{1}^{\frac{5}{2}}} \cdot \frac{1}{9}}{{x}^{2}}} \]

7. Applied rewrites 24.7%

   \[\leadsto \mathsf{fma}\left(\frac{\sqrt[3]{{x}^{4}}}{x}, \color{blue}{\frac{0.3333333333333333}{x}}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

8. Taylor expanded in x around 0

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

   1. lift-cbrt.f64 97.0

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{0.3333333333333333}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{0.3333333333333333}}{x}, \frac{\frac{\sqrt[3]{x}}{1}}{x} \cdot \frac{0.1111111111111111}{x}\right) \]
11. Add Preprocessing

Alternative 6: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66 \cdot 10^{+155}:\\ \;\;\;\;\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.66e+155)
   (* (cbrt (pow x -2.0)) 0.3333333333333333)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.66e+155) {
		tmp = cbrt(pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.66e+155) {
		tmp = Math.cbrt(Math.pow(x, -2.0)) * 0.3333333333333333;
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.66e+155)
		tmp = Float64(cbrt((x ^ -2.0)) * 0.3333333333333333);
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.66e+155], N[(N[Power[N[Power[x, -2.0], $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 < 1.6600000000000001e155

   1. Initial program 9.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 94.7

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

   5. Applied rewrites 94.7%

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

   if 1.6600000000000001e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. 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 7.0

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

   5. Applied rewrites 7.0%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. 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 51.7

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

5. Applied rewrites 51.7%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
6. 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. pow2 N/A

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

   8. cbrt-prod N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-cbrt.f64 96.4

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

7. Applied rewrites 96.4%

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

   1. pow1/3 96.4

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. pow-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-cbrt.f64 96.4

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

9. Applied rewrites 96.4%

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

Alternative 8: 92.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.4%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 94.7

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

   5. Applied rewrites 94.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. pow-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-log.f64 88.7

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

   7. Applied rewrites 88.7%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-prod N/A

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

      6. pow-to-exp N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. pow1/3 N/A

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

      10. cbrt-div N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 94.9

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

   9. Applied rewrites 94.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. 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 7.7

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

   5. Applied rewrites 7.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 9: 92.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.4%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 94.7

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. 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. pow2 N/A

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

      6. lift-*.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   4. 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 7.7

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

   5. Applied rewrites 7.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 10: 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. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. 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 51.7

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

5. Applied rewrites 51.7%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.7

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

7. Applied rewrites 88.7%

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

Alternative 11: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
6. 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 2025089 
(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)))