2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.6%

Time: 4.9s

Alternatives: 9

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.0411522633744856, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \sqrt[3]{{x}^{-2}} \cdot 0.06172839506172839\right)\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+16)
   (/
    (fma
     -0.1111111111111111
     (cbrt x)
     (fma
      (cbrt (pow x -5.0))
      -0.0411522633744856
      (fma
       (cbrt (pow x 4.0))
       0.3333333333333333
       (* (cbrt (pow x -2.0)) 0.06172839506172839))))
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+16) {
		tmp = fma(-0.1111111111111111, cbrt(x), fma(cbrt(pow(x, -5.0)), -0.0411522633744856, fma(cbrt(pow(x, 4.0)), 0.3333333333333333, (cbrt(pow(x, -2.0)) * 0.06172839506172839)))) / (x * x);
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+16)
		tmp = Float64(fma(-0.1111111111111111, cbrt(x), fma(cbrt((x ^ -5.0)), -0.0411522633744856, fma(cbrt((x ^ 4.0)), 0.3333333333333333, Float64(cbrt((x ^ -2.0)) * 0.06172839506172839)))) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+16], N[(N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision] + N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.0411522633744856 + 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]), $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 < 5e16

   1. Initial program 46.8%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 46.7

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

   4. Applied rewrites 46.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{-10}{243} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right)}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{-10}{243} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right)}{\color{blue}{{x}^{2}}} \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.0411522633744856, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \sqrt[3]{{x}^{-2}} \cdot 0.06172839506172839\right)\right)\right)}{x \cdot x}} \]

   if 5e16 < x

   1. Initial program 4.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 23.4

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

   7. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 24.6

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

   9. Applied rewrites 24.6%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-cbrt.f64 99.2

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

   12. Applied rewrites 99.2%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.0411522633744856, \mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+16)
   (/
    (fma
     (cbrt (pow x -5.0))
     -0.0411522633744856
     (fma
      (cbrt (pow x -2.0))
      0.06172839506172839
      (fma
       (cbrt (pow x 4.0))
       0.3333333333333333
       (* -0.1111111111111111 (cbrt x)))))
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+16) {
		tmp = fma(cbrt(pow(x, -5.0)), -0.0411522633744856, fma(cbrt(pow(x, -2.0)), 0.06172839506172839, fma(cbrt(pow(x, 4.0)), 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+16)
		tmp = Float64(fma(cbrt((x ^ -5.0)), -0.0411522633744856, fma(cbrt((x ^ -2.0)), 0.06172839506172839, fma(cbrt((x ^ 4.0)), 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, 5e+16], N[(N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * -0.0411522633744856 + 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]), $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 < 5e16

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{-10}{243} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{-10}{243} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right)}{\color{blue}{{x}^{2}}} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, -0.0411522633744856, \mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)\right)}{x \cdot x}} \]

   if 5e16 < x

   1. Initial program 4.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 23.4

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

   7. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 24.6

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

   9. Applied rewrites 24.6%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-cbrt.f64 99.2

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

   12. Applied rewrites 99.2%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3e+72)
   (/
    (/
     (fma
      (cbrt (pow x -2.0))
      0.06172839506172839
      (fma
       (cbrt (pow x 4.0))
       0.3333333333333333
       (* -0.1111111111111111 (cbrt x))))
     x)
    x)
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3e+72) {
		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 = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3e+72)
		tmp = Float64(Float64(fma(cbrt((x ^ -2.0)), 0.06172839506172839, fma(cbrt((x ^ 4.0)), 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, 3e+72], N[(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] / 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 < 3.00000000000000003e72

   1. Initial program 14.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   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 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 99.1%

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

   if 3.00000000000000003e72 < x

   1. Initial program 4.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 2.7

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

   4. Applied rewrites 2.7%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 3.4

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

   7. Applied rewrites 3.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 4.9

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

   9. Applied rewrites 4.9%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-cbrt.f64 99.2

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

   12. Applied rewrites 99.2%

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

Alternative 4: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right), x, 0.06172839506172839 \cdot \sqrt[3]{x}\right)}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+41)
   (/
    (/
     (fma
      (fma
       (cbrt (pow x 4.0))
       0.3333333333333333
       (* -0.1111111111111111 (cbrt x)))
      x
      (* 0.06172839506172839 (cbrt x)))
     x)
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+41) {
		tmp = (fma(fma(cbrt(pow(x, 4.0)), 0.3333333333333333, (-0.1111111111111111 * cbrt(x))), x, (0.06172839506172839 * cbrt(x))) / x) / (x * x);
	} else {
		tmp = (0.3333333333333333 * cbrt(x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+41)
		tmp = Float64(Float64(fma(fma(cbrt((x ^ 4.0)), 0.3333333333333333, Float64(-0.1111111111111111 * cbrt(x))), x, Float64(0.06172839506172839 * cbrt(x))) / x) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 * cbrt(x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+41], N[(N[(N[(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] * x + N[(0.06172839506172839 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $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 < 5.00000000000000022e41

   1. Initial program 22.2%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   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 99.0%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 99.1%

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

   if 5.00000000000000022e41 < x

   1. Initial program 4.3%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 2.9

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

   4. Applied rewrites 2.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 15.7

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

   7. Applied rewrites 15.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 17.1

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

   9. Applied rewrites 17.1%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-cbrt.f64 99.2

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

   12. Applied rewrites 99.2%

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

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+16)
   (/
    (fma
     (pow x -0.6666666666666666)
     0.06172839506172839
     (fma
      (cbrt (pow x 4.0))
      0.3333333333333333
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (/ (* 0.3333333333333333 (cbrt x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+16) {
		tmp = fma(pow(x, -0.6666666666666666), 0.06172839506172839, fma(cbrt(pow(x, 4.0)), 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+16)
		tmp = Float64(fma((x ^ -0.6666666666666666), 0.06172839506172839, fma(cbrt((x ^ 4.0)), 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, 5e+16], N[(N[(N[Power[x, -0.6666666666666666], $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[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e16

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   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 98.8%

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

      1. lift-cbrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      2. pow1/3 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{-2}\right)}^{\frac{1}{3}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      3. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{-2}\right)}^{\frac{1}{3}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      4. pow-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(-2 \cdot \frac{1}{3}\right)}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-2}{3}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -2\right)}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\frac{1}{3} \cdot -2\right)}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      8. metadata-eval 98.8

         \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   7. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{fma}\left({x}^{-0.6666666666666666}, 0.06172839506172839, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   if 5e16 < x

   1. Initial program 4.2%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 23.4

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

   7. Applied rewrites 23.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 24.6

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

   9. Applied rewrites 24.6%

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

   10. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lift-cbrt.f64 99.2

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

   12. Applied rewrites 99.2%

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

Alternative 6: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.28 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.28e+14)
   (/
    (fma
     (cbrt (pow x -2.0))
     0.06172839506172839
     (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 <= 1.28e+14) {
		tmp = fma(cbrt(pow(x, -2.0)), 0.06172839506172839, 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 <= 1.28e+14)
		tmp = Float64(fma(cbrt((x ^ -2.0)), 0.06172839506172839, 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, 1.28e+14], N[(N[(N[Power[N[Power[x, -2.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(N[Power[x, 1.3333333333333333], $MachinePrecision] * 0.3333333333333333 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $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 < 1.28e14

   1. Initial program 54.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   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 98.7%

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

      1. lift-cbrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      3. pow1/3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      4. pow-pow N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, \frac{5}{81}, \mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      6. metadata-eval 94.8

         \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   7. Applied rewrites 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{-2}}, 0.06172839506172839, \mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   if 1.28e14 < x

   1. Initial program 4.3%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. lower--.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. pow2 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. lift-*.f64 24.4

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

   7. Applied rewrites 24.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
   8. 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 25.6

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

   9. Applied rewrites 25.6%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{x}}{x} \]
   11. 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

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

   12. 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: 97.1% 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 7.0%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. flip-+ N/A

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

   4. cbrt-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower-cbrt.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. lower--.f64 5.9

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

4. Applied rewrites 5.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-fma.f64 N/A

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

   9. pow2 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. lift-*.f64 28.0

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

7. Applied rewrites 28.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]
8. 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 29.1

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

9. Applied rewrites 29.1%

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

10. Taylor expanded in x around inf

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

    1. lower-*.f64 N/A

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

    2. lift-cbrt.f64 97.3

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

12. Applied rewrites 97.3%

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

Alternative 8: 88.8% 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 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 52.5

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

5. Applied rewrites 52.5%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
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 88.9

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

7. Applied rewrites 88.9%

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

Alternative 9: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2025064 
(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)))