2cbrt (problem 3.3.4)

Percentage Accurate: 7.2% → 99.1%

Time: 3.8s

Alternatives: 10

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.2% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{\left(1 + x\right) \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1e+15)
   (/
    (- (+ 1.0 x) x)
    (+
     (pow (cbrt (+ 1.0 x)) 2.0)
     (fma (cbrt x) (cbrt x) (cbrt (* (+ 1.0 x) x)))))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.1e+15) {
		tmp = ((1.0 + x) - x) / (pow(cbrt((1.0 + x)), 2.0) + fma(cbrt(x), cbrt(x), cbrt(((1.0 + x) * x))));
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1e+15)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((cbrt(Float64(1.0 + x)) ^ 2.0) + fma(cbrt(x), cbrt(x), cbrt(Float64(Float64(1.0 + x) * x)))));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.1e+15], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1e15

   1. Initial program 54.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-pow.f64 52.4

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

   4. Applied rewrites 52.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow1/3 N/A

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

      6. flip3-- N/A

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

      7. lower-/.f64 N/A

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

      8. rem-cube-cbrt N/A

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

      9. rem-cube-cbrt N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lift-cbrt.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

   6. Applied rewrites 98.9%

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

   if 2.1e15 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 22.3

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-cbrt.f64 99.0

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

   10. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \mathbf{if}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{{t\_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (if (<= x 2.1e+15)
     (/ (- (- x -1.0) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0))))
     (/ (* 0.3333333333333333 (cbrt x)) x))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	double tmp;
	if (x <= 2.1e+15) {
		tmp = ((x - -1.0) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x - -1.0));
	double tmp;
	if (x <= 2.1e+15) {
		tmp = ((x - -1.0) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	} else {
		tmp = (0.3333333333333333 * Math.cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	tmp = 0.0
	if (x <= 2.1e+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(0.3333333333333333 * cbrt(x)) / 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, 2.1e+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[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1e15

   1. Initial program 54.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 98.5%

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

   if 2.1e15 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.1

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

   5. Applied rewrites 21.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 22.3

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

   7. Applied rewrites 22.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-cbrt.f64 99.0

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

   10. Applied rewrites 99.0%

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

Alternative 3: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{\frac{0.3333333333333333}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (cbrt (pow x -5.0))
  -0.1111111111111111
  (/ (/ 0.3333333333333333 (cbrt x)) (cbrt x))))

C

double code(double x) {
	return fma(cbrt(pow(x, -5.0)), -0.1111111111111111, ((0.3333333333333333 / cbrt(x)) / cbrt(x)));
}

Julia

function code(x)
	return fma(cbrt((x ^ -5.0)), -0.1111111111111111, Float64(Float64(0.3333333333333333 / cbrt(x)) / cbrt(x)))
end

Wolfram

code[x_] := N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.1111111111111111 + N[(N[(0.3333333333333333 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 26.9

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

5. Applied rewrites 26.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 28.0

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

7. Applied rewrites 28.0%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-flip N/A

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

   11. metadata-eval N/A

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

   12. pow-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow1/3 N/A

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

   17. lower-pow.f64 N/A

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

   18. lift-cbrt.f64 97.2

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]

10. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. pow1/3 N/A

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

    5. pow-pow N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. pow-pow N/A

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

    9. inv-pow N/A

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

    10. metadata-eval N/A

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

    11. pow-pow N/A

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

    12. pow1/3 N/A

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

    13. inv-pow N/A

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

    14. lift-pow.f64 N/A

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

    15. lift-cbrt.f64 N/A

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

    16. pow2 N/A

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

    17. associate-*l* N/A

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

    18. lift-cbrt.f64 N/A

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

    19. lift-pow.f64 N/A

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

    20. inv-pow N/A

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

    21. cbrt-div N/A

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

    22. metadata-eval N/A

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

    23. associate-*l/ N/A

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

    24. lower-/.f64 N/A

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

12. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{1 \cdot \frac{0.3333333333333333}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right) \]

13. Final simplification 97.2%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, \frac{\frac{0.3333333333333333}{\sqrt[3]{x}}}{\sqrt[3]{x}}\right) \]
14. Add Preprocessing

Alternative 4: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(cbrt(x), -2.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((cbrt(x) ^ -2.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 26.9

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

5. Applied rewrites 26.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 28.0

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

7. Applied rewrites 28.0%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/3 N/A

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

   10. pow-flip N/A

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

   11. metadata-eval N/A

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

   12. pow-pow N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. pow-pow N/A

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

   16. pow1/3 N/A

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

   17. lower-pow.f64 N/A

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

   18. lift-cbrt.f64 97.2

       \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]

10. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, \color{blue}{-0.1111111111111111}, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]
11. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. pow1/3 N/A

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

    3. lift-pow.f64 N/A

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

    4. pow-pow N/A

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

    5. lower-pow.f64 N/A

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

    6. metadata-eval 97.2

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]

12. Applied rewrites 97.2%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]
13. Add Preprocessing

Alternative 5: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+49)
   (/
    (/
     (fma
      (cbrt (* (* x x) (* x x)))
      0.3333333333333333
      (* -0.1111111111111111 (cbrt x)))
     x)
    x)
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+49) {
		tmp = (fma(cbrt(((x * x) * (x * x))), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))) / x) / x;
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+49)
		tmp = Float64(Float64(fma(cbrt(Float64(Float64(x * x) * Float64(x * x))), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))) / x) / x);
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+49], N[(N[(N[(N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000004e49

   1. Initial program 27.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 92.9

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

   7. Applied rewrites 92.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 93.0

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

   9. Applied rewrites 93.0%

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

   if 5.0000000000000004e49 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 11.3

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

   5. Applied rewrites 11.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-cbrt.f64 99.1

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

   10. Applied rewrites 99.1%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2e+47)
   (/
    (fma
     (cbrt (* (* x x) (* x x)))
     0.3333333333333333
     (* -0.1111111111111111 (cbrt x)))
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 4.2e+47) {
		tmp = fma(cbrt(((x * x) * (x * x))), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))) / (x * x);
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2e+47)
		tmp = Float64(fma(cbrt(Float64(Float64(x * x) * Float64(x * x))), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.2e+47], N[(N[(N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e47

   1. Initial program 29.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

   if 4.2e47 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 12.2

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

   5. Applied rewrites 12.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 13.5

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

   7. Applied rewrites 13.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-cbrt.f64 99.1

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

   10. Applied rewrites 99.1%

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

Alternative 7: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+14)
   (/
    (fma
     (pow x 1.3333333333333333)
     0.3333333333333333
     (* -0.1111111111111111 (cbrt x)))
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+14) {
		tmp = fma(pow(x, 1.3333333333333333), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))) / (x * x);
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+14)
		tmp = Float64(fma((x ^ 1.3333333333333333), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+14], N[(N[(N[Power[x, 1.3333333333333333], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e14

   1. Initial program 57.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 82.5

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

   7. Applied rewrites 82.5%

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

   if 2e14 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.8

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

   5. Applied rewrites 21.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 22.9

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

   7. Applied rewrites 22.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-cbrt.f64 99.0

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

   10. Applied rewrites 99.0%

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

Alternative 8: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.3333333333333333 (cbrt x)) x))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cbrt.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 26.9

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

5. Applied rewrites 26.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-fma.f64 28.0

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

7. Applied rewrites 28.0%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lift-cbrt.f64 95.9

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

10. Applied rewrites 95.9%

    \[\leadsto \frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x} \]
11. Add Preprocessing

Alternative 9: 88.6% 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 8.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 53.9

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

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
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 88.0

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

7. Applied rewrites 88.0%

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

Alternative 10: 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 8.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2025040 
(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)))