2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.1%

Time: 8.9s

Alternatives: 16

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: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. pow-sqr N/A

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

   7. lower-cbrt.f64 N/A

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

   8. pow-sqr N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. lower-*.f64 N/A

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

   13. unpow3 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cbrt.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 22.4

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

5. Applied rewrites 22.4%

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

7. Applied rewrites 49.8%

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

9. Applied rewrites 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (*
    (* (fma 0.3333333333333333 x -0.1111111111111111) (cbrt x))
    (/ 1.0 (* x x)))
   (* (/ 1.0 (sqrt x)) (* (pow x -0.16666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) * (1.0 / (x * x));
	} else {
		tmp = (1.0 / sqrt(x)) * (pow(x, -0.16666666666666666) * 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) * Float64(1.0 / Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 / sqrt(x)) * Float64((x ^ -0.16666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -0.16666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.6%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{1}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}\right) \cdot \frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (/ (* (fma 0.3333333333333333 x -0.1111111111111111) (cbrt x)) (* x x))
   (* (/ 1.0 (sqrt x)) (* (pow x -0.16666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) / (x * x);
	} else {
		tmp = (1.0 / sqrt(x)) * (pow(x, -0.16666666666666666) * 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) / Float64(x * x));
	else
		tmp = Float64(Float64(1.0 / sqrt(x)) * Float64((x ^ -0.16666666666666666) * 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, -0.16666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right) \cdot \color{blue}{\frac{1}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left({x}^{-0.16666666666666666} \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (/ (* (fma 0.3333333333333333 x -0.1111111111111111) (cbrt x)) (* x x))
   (* (/ 0.3333333333333333 (sqrt x)) (pow x -0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) / (x * x);
	} else {
		tmp = (0.3333333333333333 / sqrt(x)) * pow(x, -0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) * cbrt(x)) / Float64(x * x));
	else
		tmp = Float64(Float64(0.3333333333333333 / sqrt(x)) * (x ^ -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.16666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto {x}^{-0.16666666666666666} \cdot \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x \cdot x} \cdot \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (* (/ (fma 0.3333333333333333 x -0.1111111111111111) (* x x)) (cbrt x))
   (* (/ 0.3333333333333333 (sqrt x)) (pow x -0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) / (x * x)) * cbrt(x);
	} else {
		tmp = (0.3333333333333333 / sqrt(x)) * pow(x, -0.16666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) / Float64(x * x)) * cbrt(x));
	else
		tmp = Float64(Float64(0.3333333333333333 / sqrt(x)) * (x ^ -0.16666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.16666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 97.5%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto {x}^{-0.16666666666666666} \cdot \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x \cdot x} \cdot \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x \cdot x} \cdot \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (* (/ (fma 0.3333333333333333 x -0.1111111111111111) (* x x)) (cbrt x))
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) / (x * x)) * cbrt(x);
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) / Float64(x * x)) * cbrt(x));
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   9. Applied rewrites 97.5%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-0.16666666666666666}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x \cdot x} \cdot \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (/ (* (* 0.3333333333333333 x) (cbrt x)) (* x x))
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = ((0.3333333333333333 * x) * cbrt(x)) / (x * x);
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = ((0.3333333333333333 * x) * Math.cbrt(x)) / (x * x);
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) / Math.sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(Float64(0.3333333333333333 * x) * cbrt(x)) / Float64(x * x));
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.0%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

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

   9. Applied rewrites 92.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-0.16666666666666666}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. metadata-eval N/A

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

   6. pow-sqr N/A

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

   7. lower-cbrt.f64 N/A

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

   8. pow-sqr N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. lower-*.f64 N/A

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

   13. unpow3 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cbrt.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 22.4

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

5. Applied rewrites 22.4%

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

7. Applied rewrites 49.8%

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

9. Applied rewrites 98.2%

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

Alternative 9: 92.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x} \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (/ (* (* 0.3333333333333333 x) (cbrt x)) (* x x))
   (*
    (/ (fma 0.3333333333333333 x -0.1111111111111111) x)
    (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = ((0.3333333333333333 * x) * cbrt(x)) / (x * x);
	} else {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) / x) * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(Float64(Float64(0.3333333333333333 * x) * cbrt(x)) / Float64(x * x));
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) / x) * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision] * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 96.0%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 2.1%

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

   9. Applied rewrites 89.2%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x\right) \cdot \sqrt[3]{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x} \cdot {x}^{-0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x} \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (*
    (/ (fma 0.3333333333333333 x -0.1111111111111111) x)
    (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (fma(0.3333333333333333, x, -0.1111111111111111) / x) * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.1111111111111111) / x) * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision] / x), $MachinePrecision] * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-sqr N/A

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

      7. lower-cbrt.f64 N/A

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

      8. pow-sqr N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cbrt.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 0.0

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

   5. Applied rewrites 0.0%

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

   7. Applied rewrites 2.1%

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

   9. Applied rewrites 89.2%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)}{x} \cdot {x}^{-0.6666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.34e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (/ 1.0 (pow x 0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.34e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / Math.pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.34e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / (x ^ 0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.34e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.34000000000000001e154

   1. Initial program 7.8%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.9

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

   5. Applied rewrites 95.9%

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

   if 1.34000000000000001e154 < x

   1. Initial program 4.7%

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   5. Applied rewrites 4.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.2%

      \[\leadsto \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 1.0 (pow x 0.6666666666666666)) 0.3333333333333333))

C

double code(double x) {
	return (1.0 / pow(x, 0.6666666666666666)) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x ** 0.6666666666666666d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return (1.0 / Math.pow(x, 0.6666666666666666)) * 0.3333333333333333;
}

Python

def code(x):
	return (1.0 / math.pow(x, 0.6666666666666666)) * 0.3333333333333333

Julia

function code(x)
	return Float64(Float64(1.0 / (x ^ 0.6666666666666666)) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x ^ 0.6666666666666666)) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 50.3

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

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

7. Applied rewrites 89.2%

   \[\leadsto \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 13: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (sqrt x) -1.3333333333333333) 0.3333333333333333))

C

double code(double x) {
	return pow(sqrt(x), -1.3333333333333333) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt(x) ** (-1.3333333333333333d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(x), -1.3333333333333333) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(math.sqrt(x), -1.3333333333333333) * 0.3333333333333333

Julia

function code(x)
	return Float64((sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.3333333333333333], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 50.3

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

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

7. Applied rewrites 89.2%

   \[\leadsto {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 14: 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

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.6666666666666666d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(x, -0.6666666666666666) * 0.3333333333333333

Julia

function code(x)
	return Float64((x ^ -0.6666666666666666) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.6666666666666666) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 50.3

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

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

7. Applied rewrites 89.2%

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

Alternative 15: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

code[x_] := N[Power[x, 1/3], $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. rem-exp-log N/A

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

   2. lift-cbrt.f64 N/A

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

   3. pow1/3 N/A

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

   4. log-pow N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-log.f64 6.5

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

4. Applied rewrites 6.5%

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

5. Taylor expanded in x around inf

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

   1. lower-cbrt.f64 5.2

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

7. Applied rewrites 5.2%

   \[\leadsto \color{blue}{\sqrt[3]{x}} \]
8. Add Preprocessing

Alternative 16: 4.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.3%

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

   1. rem-exp-log N/A

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

   2. lift-cbrt.f64 N/A

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

   3. pow1/3 N/A

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

   4. log-pow N/A

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

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-log.f64 6.5

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

4. Applied rewrites 6.5%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto x \cdot \left(\color{blue}{0} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) \]

   3. mul0-lft N/A

      \[\leadsto x \cdot \color{blue}{0} \]

   4. mul0-rgt 4.2

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 4.2%

   \[\leadsto \color{blue}{0} \]
8. 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 2024240 
(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)))