2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 99.0%

Time: 10.6s

Alternatives: 9

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(x + 1\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.9e+14)
   (/
    (- (+ x 1.0) x)
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (pow (+ x 1.0) 0.6666666666666666)))
   (/ (* (cbrt x) 0.3333333333333333) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.9e+14) {
		tmp = ((x + 1.0) - x) / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), pow((x + 1.0), 0.6666666666666666));
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.9e+14)
		tmp = Float64(Float64(Float64(x + 1.0) - x) / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), (Float64(x + 1.0) ^ 0.6666666666666666)));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.9e+14], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9e14

   1. Initial program 57.9%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

      3. metadata-eval N/A

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

      4. difference-of-sqr-1 N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

      9. sub-neg N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. sub-neg N/A

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

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

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

      15. metadata-eval 58.3

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

   4. Applied egg-rr 58.3%

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

      1. +-lft-identity N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. sqr-pow N/A

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

      5. unpow-prod-down N/A

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

      6. sqr-neg N/A

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

      7. unpow-prod-down N/A

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

      8. sqr-pow N/A

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

      9. cube-neg N/A

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

      10. rem-cube-cbrt N/A

          \[\leadsto \sqrt[3]{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \frac{1}{x + -1}\right)} - \frac{0 + \left(\mathsf{neg}\left(\color{blue}{x}\right)\right)}{0 \cdot 0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - 0 \cdot \sqrt[3]{x}\right)} \]

      11. sub-neg N/A

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

      12. neg-sub0 N/A

          \[\leadsto \sqrt[3]{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \frac{1}{x + -1}\right)} - \frac{\color{blue}{\mathsf{neg}\left(x\right)}}{0 \cdot 0 + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - 0 \cdot \sqrt[3]{x}\right)} \]

      13. rem-cube-cbrt N/A

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

      14. cube-neg N/A

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

      15. sqr-pow N/A

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

      16. unpow-prod-down N/A

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

      17. sqr-neg N/A

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

      18. unpow-prod-down N/A

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

      19. sqr-pow N/A

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

      20. rem-cube-cbrt N/A

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

   6. Applied egg-rr 54.6%

      \[\leadsto \sqrt[3]{\left(x + 1\right) \cdot \left(\left(x + -1\right) \cdot \frac{1}{x + -1}\right)} - \color{blue}{\frac{x}{0 + \left({x}^{0.6666666666666666} - 0\right)}} \]

   8. Applied egg-rr 98.2%

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

   if 3.9e14 < x

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 46.9

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

   5. Simplified 46.9%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 98.4

         \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-flip N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. +-lft-identity N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

      10. flip-+ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

      11. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. un-div-inv N/A

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

       4. associate-*l/ N/A

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

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

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

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

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

       7. cbrt-lowering-cbrt.f64 99.2

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

   11. Applied egg-rr 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(x + 1\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \mathsf{fma}\left(0.06172839506172839, \sqrt[3]{\frac{1}{x \cdot x}}, \sqrt[3]{x} \cdot -0.1111111111111111\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+15)
   (/
    (fma
     0.3333333333333333
     (cbrt (* (* x x) (* x x)))
     (fma
      0.06172839506172839
      (cbrt (/ 1.0 (* x x)))
      (* (cbrt x) -0.1111111111111111)))
    (* x x))
   (/ (* (cbrt x) 0.3333333333333333) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+15) {
		tmp = fma(0.3333333333333333, cbrt(((x * x) * (x * x))), fma(0.06172839506172839, cbrt((1.0 / (x * x))), (cbrt(x) * -0.1111111111111111))) / (x * x);
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(fma(0.3333333333333333, cbrt(Float64(Float64(x * x) * Float64(x * x))), fma(0.06172839506172839, cbrt(Float64(1.0 / Float64(x * x))), Float64(cbrt(x) * -0.1111111111111111))) / Float64(x * x));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+15], N[(N[(0.3333333333333333 * N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(0.06172839506172839 * N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e15

   1. Initial program 57.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.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. Simplified 87.2%

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

   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

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 46.9

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

   5. Simplified 46.9%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 98.4

         \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-flip N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. +-lft-identity N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

      10. flip-+ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

      11. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. un-div-inv N/A

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

       4. associate-*l/ N/A

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

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

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

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

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

       7. cbrt-lowering-cbrt.f64 99.2

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

   11. Applied egg-rr 99.2%

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

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+15)
   (/
    (fma
     0.3333333333333333
     (cbrt (* (* x x) (* x x)))
     (* (cbrt x) -0.1111111111111111))
    (* x x))
   (/ (* (cbrt x) 0.3333333333333333) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+15) {
		tmp = fma(0.3333333333333333, cbrt(((x * x) * (x * x))), (cbrt(x) * -0.1111111111111111)) / (x * x);
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+15)
		tmp = Float64(fma(0.3333333333333333, cbrt(Float64(Float64(x * x) * Float64(x * x))), Float64(cbrt(x) * -0.1111111111111111)) / Float64(x * x));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+15], N[(N[(0.3333333333333333 * N[Power[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e15

   1. Initial program 57.9%

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

   3. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. pow-sqr N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 84.3

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

   5. Simplified 84.3%

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

   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

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 46.9

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

   5. Simplified 46.9%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 98.4

         \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

   7. Applied egg-rr 98.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. pow-flip N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. +-lft-identity N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

      10. flip-+ N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

      11. neg-sub0 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

      12. distribute-neg-frac2 N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

      13. distribute-neg-frac N/A

          \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

   9. Applied egg-rr 99.0%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. *-commutative N/A

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

       3. un-div-inv N/A

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

       4. associate-*l/ N/A

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

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

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

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

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

       7. cbrt-lowering-cbrt.f64 99.2

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

   11. Applied egg-rr 99.2%

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

Alternative 4: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.0

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

5. Simplified 48.0%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 95.5

      \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

7. Applied egg-rr 95.5%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow2 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-frac-neg2 N/A

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

   8. remove-double-neg N/A

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

   9. +-lft-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

   10. flip-+ N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

   11. neg-sub0 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

   13. distribute-neg-frac N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

9. Applied egg-rr 96.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. *-commutative N/A

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

    3. un-div-inv N/A

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

    4. associate-*l/ N/A

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

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

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

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

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

    7. cbrt-lowering-cbrt.f64 96.2

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

11. Applied egg-rr 96.2%

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

Alternative 5: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.0

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

5. Simplified 48.0%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 95.5

      \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

7. Applied egg-rr 95.5%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow2 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-frac-neg2 N/A

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

   8. remove-double-neg N/A

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

   9. +-lft-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

   10. flip-+ N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

   11. neg-sub0 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

   13. distribute-neg-frac N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

9. Applied egg-rr 96.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
10. Step-by-step derivation

    1. associate-*r* N/A

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

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

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

    3. un-div-inv N/A

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

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

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

    5. cbrt-lowering-cbrt.f64 96.1

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

11. Applied egg-rr 96.1%

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

12. Final simplification 96.1%

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

Alternative 6: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.0

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

5. Simplified 48.0%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 95.5

      \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}} \]

7. Applied egg-rr 95.5%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

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

   2. pow-flip N/A

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

   3. pow2 N/A

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

   4. associate-/r* N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-frac-neg2 N/A

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

   8. remove-double-neg N/A

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

   9. +-lft-identity N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{0 + \sqrt[3]{x}}} \]

   10. flip-+ N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{0 - \sqrt[3]{x}}}} \]

   11. neg-sub0 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)}}} \]

   12. distribute-neg-frac2 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\mathsf{neg}\left(\frac{0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x}}\right)}} \]

   13. distribute-neg-frac N/A

       \[\leadsto \frac{1}{3} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{\color{blue}{\frac{\mathsf{neg}\left(\left(0 \cdot 0 - \sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}{\sqrt[3]{x}}}} \]

9. Applied egg-rr 96.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\frac{1}{x} \cdot \sqrt[3]{x}\right)} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. un-div-inv N/A

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

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

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

    4. cbrt-lowering-cbrt.f64 96.1

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

11. Applied egg-rr 96.1%

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

Alternative 7: 88.8% accurate, 1.8× 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 8.4%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.0

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

5. Simplified 48.0%

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

   1. pow1/3 N/A

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

   2. inv-pow N/A

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

   3. pow2 N/A

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

   4. pow-pow N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-flip N/A

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

   10. un-div-inv N/A

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

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

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

   12. pow-lowering-pow.f64 88.0

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

7. Applied egg-rr 88.0%

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

Alternative 8: 88.8% accurate, 1.9× 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 8.4%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.0

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

5. Simplified 48.0%

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

   1. *-commutative N/A

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

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

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

   3. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. pow2 N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

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

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

   10. metadata-eval 88.0

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

7. Applied egg-rr 88.0%

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

8. Final simplification 88.0%

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

Alternative 9: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

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

   2. cbrt-lowering-cbrt.f64 1.8

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

5. Simplified 1.8%

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

6. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

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

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   3. cbrt-lowering-cbrt.f64 1.8

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

8. Simplified 1.8%

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

   1. +-lft-identity N/A

      \[\leadsto \color{blue}{0 + \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)} \]

   2. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} + {\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)}^{3}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)}} \]

   3. sqr-pow N/A

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

   4. unpow-prod-down N/A

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

   5. sqr-neg N/A

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

   6. unpow-prod-down N/A

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

   7. sqr-pow N/A

      \[\leadsto \frac{{0}^{3} + \color{blue}{{\left(\sqrt[3]{x}\right)}^{3}}}{0 \cdot 0 + \left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right) - 0 \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)} \]

   8. sqr-neg N/A

      \[\leadsto \frac{{0}^{3} + {\left(\sqrt[3]{x}\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}} - 0 \cdot \left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right)\right)} \]

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. neg-mul-1 N/A

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

   12. associate-*r* N/A

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

   13. metadata-eval N/A

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

   14. associate-*r* N/A

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

   15. metadata-eval N/A

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

10. Applied egg-rr 5.5%

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

Developer Target 1: 98.4% 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 2024205 
(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)))