2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.1%

Time: 10.5s

Alternatives: 10

Speedup: 2.0×

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: 6.9% 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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 48000000:\\ \;\;\;\;\sqrt[3]{\left(1 - x \cdot x\right) \cdot \frac{1}{1 - x}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 48000000.0)
   (- (cbrt (* (- 1.0 (* x x)) (/ 1.0 (- 1.0 x)))) (cbrt x))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 48000000.0) {
		tmp = cbrt(((1.0 - (x * x)) * (1.0 / (1.0 - x)))) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 48000000.0) {
		tmp = Math.cbrt(((1.0 - (x * x)) * (1.0 / (1.0 - x)))) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 / (x / Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 48000000.0)
		tmp = Float64(cbrt(Float64(Float64(1.0 - Float64(x * x)) * Float64(1.0 / Float64(1.0 - x)))) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 48000000.0], N[(N[Power[N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e7

   1. Initial program 79.7%

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

      1. rem-cube-cbrt N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left({\left(\sqrt[3]{x + 1}\right)}^{3}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{3}{2}\right)}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. pow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left({\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{3}{2}\right)}\right)}^{2}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\left({\left(\sqrt[3]{x + 1}\right)}^{\left(\frac{3}{2}\right)}\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. pow1/3 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\left({\left({\left(x + 1\right)}^{\frac{1}{3}}\right)}^{\left(\frac{3}{2}\right)}\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      6. pow-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{3} \cdot \frac{3}{2}\right)}\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{1}{3} \cdot \frac{3}{2}\right)\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{1}{3} \cdot \frac{3}{2}\right)\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{1}{3} \cdot \frac{3}{2}\right)\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      10. metadata-eval 79.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right), 2\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   4. Applied egg-rr 79.2%

      \[\leadsto \sqrt[3]{\color{blue}{{\left({\left(x + 1\right)}^{0.5}\right)}^{2}}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. pow-pow N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{1}{2} \cdot 2\right)}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left({\left(x + 1\right)}^{1}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      3. unpow1 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(x + 1\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(1 + x\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      5. flip-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1 \cdot 1 - x \cdot x}{1 - x}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\left(\left(1 \cdot 1 - x \cdot x\right) \cdot \frac{1}{1 - x}\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(1 \cdot 1 - x \cdot x\right), \left(\frac{1}{1 - x}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\left(1 - x \cdot x\right), \left(\frac{1}{1 - x}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot x\right)\right), \left(\frac{1}{1 - x}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{1}{1 - x}\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, \left(1 - x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

      12. --lowering--.f64 80.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(x\right)\right) \]

   6. Applied egg-rr 80.2%

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

   if 4.8e7 < x

   1. Initial program 5.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 50.4%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

      10. pow-lowering-pow.f64 89.9%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

   7. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{-1}{3} + \color{blue}{1}\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right)}\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot \color{blue}{x}\right)\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\frac{1}{3}}} \cdot x\right)\right) \]

      5. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{x}} \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\sqrt[3]{x}} \cdot x\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{x}} \cdot x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right)\right) \]

      9. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{1}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{x}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right) \]

      13. cbrt-lowering-cbrt.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \mathsf{cbrt.f64}\left(x\right)\right)\right) \]

   9. Applied egg-rr 98.6%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{x \cdot \left(\frac{1}{\sqrt[3]{\frac{1}{\frac{1}{x} + \frac{2}{x \cdot x}}}} + \sqrt[3]{\frac{1}{x}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  0.6666666666666666
  (*
   x
   (+ (/ 1.0 (cbrt (/ 1.0 (+ (/ 1.0 x) (/ 2.0 (* x x)))))) (cbrt (/ 1.0 x))))))

C

double code(double x) {
	return 0.6666666666666666 / (x * ((1.0 / cbrt((1.0 / ((1.0 / x) + (2.0 / (x * x)))))) + cbrt((1.0 / x))));
}

Java

public static double code(double x) {
	return 0.6666666666666666 / (x * ((1.0 / Math.cbrt((1.0 / ((1.0 / x) + (2.0 / (x * x)))))) + Math.cbrt((1.0 / x))));
}

Julia

function code(x)
	return Float64(0.6666666666666666 / Float64(x * Float64(Float64(1.0 / cbrt(Float64(1.0 / Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))))) + cbrt(Float64(1.0 / x)))))
end

Wolfram

code[x_] := N[(0.6666666666666666 / N[(x * N[(N[(1.0 / N[Power[N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right), \color{blue}{\left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\right) \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \color{blue}{\left(x \cdot \left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}}\right), \color{blue}{\left(\sqrt[3]{\frac{1}{x}}\right)}\right)\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{1}{x}}}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{\color{blue}{1}}{x}}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2 \cdot 1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left({x}^{2}\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   12. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 97.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

7. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}\right)}} \]
8. Step-by-step derivation

   1. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}{\frac{1}{x} - \frac{2}{x \cdot x}}}\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, x\right)\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{\frac{\frac{1}{x} - \frac{2}{x \cdot x}}{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}}}\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, x\right)\right)\right)\right)\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\frac{1}{x} - \frac{2}{x \cdot x}}{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}}}\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, x\right)}\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{\sqrt[3]{\frac{\frac{1}{x} - \frac{2}{x \cdot x}}{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}}}\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, x\right)\right)\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt[3]{\frac{\frac{1}{x} - \frac{2}{x \cdot x}}{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}}\right)\right), \mathsf{cbrt.f64}\left(\color{blue}{\mathsf{/.f64}\left(1, x\right)}\right)\right)\right)\right) \]

   6. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{\frac{1}{x} - \frac{2}{x \cdot x}}{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{x}\right)\right)\right)\right)\right) \]

   7. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{\frac{1}{x} \cdot \frac{1}{x} - \frac{2}{x \cdot x} \cdot \frac{2}{x \cdot x}}{\frac{1}{x} - \frac{2}{x \cdot x}}}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\left(\frac{1}{\frac{1}{x} + \frac{2}{x \cdot x}}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{x} + \frac{2}{x \cdot x}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(\frac{2}{x \cdot x}\right)\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

9. Applied egg-rr 97.6%

   \[\leadsto \frac{0.6666666666666666}{x \cdot \left(\color{blue}{\frac{1}{\sqrt[3]{\frac{1}{\frac{1}{x} + \frac{2}{x \cdot x}}}}} + \sqrt[3]{\frac{1}{x}}\right)} \]
10. Add Preprocessing

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \frac{1}{\sqrt[3]{x}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  0.6666666666666666
  (* x (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x)))) (/ 1.0 (cbrt x))))))

C

double code(double x) {
	return 0.6666666666666666 / (x * (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / cbrt(x))));
}

Java

public static double code(double x) {
	return 0.6666666666666666 / (x * (Math.cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / Math.cbrt(x))));
}

Julia

function code(x)
	return Float64(0.6666666666666666 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(1.0 / cbrt(x)))))
end

Wolfram

code[x_] := N[(0.6666666666666666 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right), \color{blue}{\left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\right) \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \color{blue}{\left(x \cdot \left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}}\right), \color{blue}{\left(\sqrt[3]{\frac{1}{x}}\right)}\right)\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{1}{x}}}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{\color{blue}{1}}{x}}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2 \cdot 1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left({x}^{2}\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   12. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 97.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

7. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}\right)}} \]
8. Step-by-step derivation

   1. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x}}}\right)\right)\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\frac{1}{\sqrt[3]{\color{blue}{x}}}\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right)\right)\right) \]

   4. cbrt-lowering-cbrt.f64 97.6%

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{/.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right)\right)\right)\right) \]

9. Applied egg-rr 97.6%

   \[\leadsto \frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \color{blue}{\frac{1}{\sqrt[3]{x}}}\right)} \]
10. Add Preprocessing

Alternative 4: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  0.6666666666666666
  (* x (+ (cbrt (/ 1.0 x)) (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x))))))))

C

double code(double x) {
	return 0.6666666666666666 / (x * (cbrt((1.0 / x)) + cbrt(((1.0 / x) + (2.0 / (x * x))))));
}

Java

public static double code(double x) {
	return 0.6666666666666666 / (x * (Math.cbrt((1.0 / x)) + Math.cbrt(((1.0 / x) + (2.0 / (x * x))))));
}

Julia

function code(x)
	return Float64(0.6666666666666666 / Float64(x * Float64(cbrt(Float64(1.0 / x)) + cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))))))
end

Wolfram

code[x_] := N[(0.6666666666666666 / N[(x * N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.1%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right), \color{blue}{\left(\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\right) \]

4. Applied egg-rr 7.3%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \color{blue}{\left(x \cdot \left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}} + \sqrt[3]{\frac{1}{x}}\right)}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}}\right), \color{blue}{\left(\sqrt[3]{\frac{1}{x}}\right)}\right)\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{x} + 2 \cdot \frac{1}{{x}^{2}}\right)\right), \left(\sqrt[3]{\color{blue}{\frac{1}{x}}}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{x}\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{\color{blue}{1}}{x}}\right)\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(2 \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2 \cdot 1}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{2}{{x}^{2}}\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left({x}^{2}\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \left(x \cdot x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \left(\sqrt[3]{\frac{1}{x}}\right)\right)\right)\right) \]

   12. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\left(\frac{1}{x}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 97.5%

       \[\leadsto \mathsf{/.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right)\right) \]

7. Simplified 97.5%

   \[\leadsto \color{blue}{\frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}\right)}} \]

8. Final simplification 97.5%

   \[\leadsto \frac{0.6666666666666666}{x \cdot \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}}\right)} \]
9. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 39000000:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 39000000.0)
   (- (cbrt (+ x 1.0)) (cbrt x))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 39000000.0) {
		tmp = cbrt((x + 1.0)) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 39000000.0) {
		tmp = Math.cbrt((x + 1.0)) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 / (x / Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 39000000.0)
		tmp = Float64(cbrt(Float64(x + 1.0)) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 39000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9e7

   1. Initial program 79.7%

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

   if 3.9e7 < x

   1. Initial program 5.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 50.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 50.4%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

      10. pow-lowering-pow.f64 89.9%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

   7. Applied egg-rr 89.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{-1}{3} + \color{blue}{1}\right)}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right)}\right)\right) \]

      3. pow-plus N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot \color{blue}{x}\right)\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\frac{1}{3}}} \cdot x\right)\right) \]

      5. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{x}} \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\sqrt[3]{x}} \cdot x\right)\right) \]

      7. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{x}} \cdot x\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right)\right) \]

      9. cbrt-div N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{1}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{x}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right) \]

      13. cbrt-lowering-cbrt.f64 98.6%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \mathsf{cbrt.f64}\left(x\right)\right)\right) \]

   9. Applied egg-rr 98.6%

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

Alternative 6: 92.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 11.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 93.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 93.1%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

      10. pow-lowering-pow.f64 86.8%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(2 \cdot \color{blue}{\frac{1}{3}}\right)}\right)\right) \]

      2. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({\left(x \cdot x\right)}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      4. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x \cdot x}\right)\right) \]

      5. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(x \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 93.3%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{*.f64}\left(x, x\right)\right)\right) \]

   9. Applied egg-rr 93.3%

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

   if 1.35000000000000003e154 < 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

      4. cbrt-lowering-cbrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

      9. *-lowering-*.f64 4.7%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

   5. Simplified 4.7%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

      6. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

      7. pow1/3 N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

      8. pow-sqr N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

      10. pow-lowering-pow.f64 89.1%

          \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

   7. Applied egg-rr 89.1%

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

Alternative 7: 97.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 49.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 49.9%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

   6. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

   7. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

   8. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

   10. pow-lowering-pow.f64 87.9%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

7. Applied egg-rr 87.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\frac{-1}{3} + \color{blue}{1}\right)}\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\left(\mathsf{neg}\left(\frac{1}{3}\right)\right) + 1\right)}\right)\right) \]

   3. pow-plus N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)} \cdot \color{blue}{x}\right)\right) \]

   4. pow-flip N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\frac{1}{3}}} \cdot x\right)\right) \]

   5. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\sqrt[3]{x}} \cdot x\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{1}}{\sqrt[3]{x}} \cdot x\right)\right) \]

   7. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{x}} \cdot x\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right)\right) \]

   9. cbrt-div N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{\sqrt[3]{1}}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(x \cdot \frac{1}{\sqrt[3]{\color{blue}{x}}}\right)\right) \]

   11. un-div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{x}{\color{blue}{\sqrt[3]{x}}}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \color{blue}{\left(\sqrt[3]{x}\right)}\right)\right) \]

   13. cbrt-lowering-cbrt.f64 96.3%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(x, \mathsf{cbrt.f64}\left(x\right)\right)\right) \]

9. Applied egg-rr 96.3%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{\frac{x}{\sqrt[3]{x}}}} \]
10. Add Preprocessing

Alternative 8: 88.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 49.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 49.9%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right) \]

   6. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{x}}\right)\right) \]

   7. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{1}{3}} \cdot {x}^{\color{blue}{\frac{1}{3}}}\right)\right) \]

   8. pow-sqr N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\color{blue}{\left(2 \cdot \frac{1}{3}\right)}}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left({x}^{\frac{2}{3}}\right)\right) \]

   10. pow-lowering-pow.f64 87.9%

       \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(x, \color{blue}{\frac{2}{3}}\right)\right) \]

7. Applied egg-rr 87.9%

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

Alternative 9: 88.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 49.9%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 49.9%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   7. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)\right), \frac{1}{3}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

   10. metadata-eval 87.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

7. Applied egg-rr 87.9%

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

8. Final simplification 87.9%

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

Alternative 10: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.1%

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

3. Taylor expanded in x around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

   2. cbrt-lowering-cbrt.f64 1.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right) \]

5. Simplified 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 2024158 
(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)))