2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 99.0%

Time: 8.2s

Alternatives: 9

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+15)
   (/
    (- (+ 1.0 x) x)
    (fma
     (cbrt x)
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (exp (* (log1p x) 0.6666666666666666))))
   (pow (/ x (* 0.3333333333333333 (cbrt x))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+15) {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), exp((log1p(x) * 0.6666666666666666)));
	} else {
		tmp = pow((x / (0.3333333333333333 * cbrt(x))), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), exp(Float64(log1p(x) * 0.6666666666666666))));
	else
		tmp = Float64(x / Float64(0.3333333333333333 * cbrt(x))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+15], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(x / N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e15

   1. Initial program 60.2%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 60.6

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

   4. Applied rewrites 60.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]

   if 2e15 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 22.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 49.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}}} \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.0%

       \[\leadsto \frac{1}{\frac{x}{0.3333333333333333 \cdot \color{blue}{\sqrt[3]{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\sqrt[3]{x \cdot x}, 3 - \frac{\frac{0.2222222222222222}{x}}{x}, \sqrt[3]{{x}^{-1}}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5e+24)
   (pow
    (fma
     (cbrt (* x x))
     (- 3.0 (/ (/ 0.2222222222222222 x) x))
     (cbrt (pow x -1.0)))
    -1.0)
   (pow (/ x (* 0.3333333333333333 (cbrt x))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2.5e+24) {
		tmp = pow(fma(cbrt((x * x)), (3.0 - ((0.2222222222222222 / x) / x)), cbrt(pow(x, -1.0))), -1.0);
	} else {
		tmp = pow((x / (0.3333333333333333 * cbrt(x))), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5e+24)
		tmp = fma(cbrt(Float64(x * x)), Float64(3.0 - Float64(Float64(0.2222222222222222 / x) / x)), cbrt((x ^ -1.0))) ^ -1.0;
	else
		tmp = Float64(x / Float64(0.3333333333333333 * cbrt(x))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.5e+24], N[Power[N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] * N[(3.0 - N[(N[(0.2222222222222222 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(x / N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000023e24

   1. Initial program 50.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 94.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 94.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 94.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x \cdot x}, \color{blue}{3 - \frac{\frac{0.2222222222222222}{x}}{x}}, \sqrt[3]{\frac{1}{x}}\right)} \]

   if 2.50000000000000023e24 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 21.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 48.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.1%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}}} \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.0%

       \[\leadsto \frac{1}{\frac{x}{0.3333333333333333 \cdot \color{blue}{\sqrt[3]{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{+24}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\sqrt[3]{x \cdot x}, 3 - \frac{\frac{0.2222222222222222}{x}}{x}, \sqrt[3]{{x}^{-1}}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+36}:\\ \;\;\;\;{\left(\frac{x}{\frac{\mathsf{fma}\left(0.06172839506172839, {x}^{-0.6666666666666666}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+36)
   (pow
    (/
     x
     (/
      (fma
       0.06172839506172839
       (pow x -0.6666666666666666)
       (* (cbrt x) (fma 0.3333333333333333 x -0.1111111111111111)))
      x))
    -1.0)
   (pow (/ x (* 0.3333333333333333 (cbrt x))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+36) {
		tmp = pow((x / (fma(0.06172839506172839, pow(x, -0.6666666666666666), (cbrt(x) * fma(0.3333333333333333, x, -0.1111111111111111))) / x)), -1.0);
	} else {
		tmp = pow((x / (0.3333333333333333 * cbrt(x))), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+36)
		tmp = Float64(x / Float64(fma(0.06172839506172839, (x ^ -0.6666666666666666), Float64(cbrt(x) * fma(0.3333333333333333, x, -0.1111111111111111))) / x)) ^ -1.0;
	else
		tmp = Float64(x / Float64(0.3333333333333333 * cbrt(x))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+36], N[Power[N[(x / N[(N[(0.06172839506172839 * N[Power[x, -0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(0.3333333333333333 * x + -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[Power[N[(x / N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e36

   1. Initial program 31.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 96.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 96.2%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 96.2%

       \[\leadsto \frac{1}{\frac{x}{\frac{\mathsf{fma}\left(0.06172839506172839, {x}^{-0.6666666666666666}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}} \]

   if 1.00000000000000004e36 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 46.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 70.6%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}}} \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 99.0%

       \[\leadsto \frac{1}{\frac{x}{0.3333333333333333 \cdot \color{blue}{\sqrt[3]{x}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+36}:\\ \;\;\;\;{\left(\frac{x}{\frac{\mathsf{fma}\left(0.06172839506172839, {x}^{-0.6666666666666666}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{0.3333333333333333 \cdot \sqrt[3]{x}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (pow (* (cbrt (* x x)) 3.0) -1.0)
   (* (pow x -0.6666666666666666) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = pow((cbrt((x * x)) * 3.0), -1.0);
	} else {
		tmp = pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.pow((Math.cbrt((x * x)) * 3.0), -1.0);
	} else {
		tmp = Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(x * x)) * 3.0) ^ -1.0;
	else
		tmp = Float64((x ^ -0.6666666666666666) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[Power[N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] * 3.0), $MachinePrecision], -1.0], $MachinePrecision], N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 10.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   7. Applied rewrites 98.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 94.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 8.6

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

   5. Applied rewrites 8.6%

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

   7. Applied rewrites 89.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;{\left(\sqrt[3]{x \cdot x} \cdot 3\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 10.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

   7. Applied rewrites 94.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 8.6

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

   5. Applied rewrites 8.6%

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

   7. Applied rewrites 89.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{\left(x \cdot x\right)}^{-1}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 26.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{\frac{1}{x}}{x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

7. Applied rewrites 52.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \mathsf{fma}\left(-0.1111111111111111, \sqrt[3]{x}, \left(\sqrt[3]{x} \cdot x\right) \cdot 0.3333333333333333\right)\right)}}} \]
8. Step-by-step derivation

9. Applied rewrites 73.6%

   \[\leadsto \frac{1}{\frac{x}{\color{blue}{\frac{\mathsf{fma}\left(0.06172839506172839, {\left(\sqrt[3]{x}\right)}^{-2}, \sqrt[3]{x} \cdot \mathsf{fma}\left(0.3333333333333333, x, -0.1111111111111111\right)\right)}{x}}}} \]

10. Taylor expanded in x around inf

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

12. Applied rewrites 96.7%

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

13. Final simplification 96.7%

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

Alternative 7: 88.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x -0.6666666666666666) 0.3333333333333333))

C

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

Fortran

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 7.5%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 53.1

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

5. Applied rewrites 53.1%

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

7. Applied rewrites 88.6%

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

Alternative 8: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 1.8

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

7. Applied rewrites 1.8%

   \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

      \[\leadsto 1 - {x}^{\color{blue}{\left(2 \cdot \frac{1}{6}\right)}} \]

   3. pow-pow N/A

      \[\leadsto 1 - \color{blue}{{\left({x}^{2}\right)}^{\frac{1}{6}}} \]

   4. pow2 N/A

      \[\leadsto 1 - {\color{blue}{\left(x \cdot x\right)}}^{\frac{1}{6}} \]

   5. sqr-neg N/A

      \[\leadsto 1 - {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\frac{1}{6}} \]

   6. lift-neg.f64 N/A

      \[\leadsto 1 - {\left(\color{blue}{\left(-x\right)} \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\frac{1}{6}} \]

   7. lift-neg.f64 N/A

      \[\leadsto 1 - {\left(\left(-x\right) \cdot \color{blue}{\left(-x\right)}\right)}^{\frac{1}{6}} \]

   8. pow-prod-down N/A

      \[\leadsto 1 - \color{blue}{{\left(-x\right)}^{\frac{1}{6}} \cdot {\left(-x\right)}^{\frac{1}{6}}} \]

   9. pow-prod-up N/A

      \[\leadsto 1 - \color{blue}{{\left(-x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}} \]

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

   12. lift-cbrt.f64 5.4

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

9. Applied rewrites 5.4%

   \[\leadsto 1 - \color{blue}{\sqrt[3]{-x}} \]
10. Add Preprocessing

Alternative 9: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
6. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024327 
(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)))