2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 99.0%

Time: 11.3s

Alternatives: 6

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: 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x} \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (* 0.3333333333333333 (/ (cbrt x) x))
     (/
      (- (+ x 1.0) x)
      (+
       (pow (+ x 1.0) 0.6666666666666666)
       (+ (pow x 0.6666666666666666) (* (cbrt x) t_0)))))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * (cbrt(x) / x);
	} else {
		tmp = ((x + 1.0) - x) / (pow((x + 1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + (cbrt(x) * t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * (Math.cbrt(x) / x);
	} else {
		tmp = ((x + 1.0) - x) / (Math.pow((x + 1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) + (Math.cbrt(x) * t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) / x));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64((Float64(x + 1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + Float64(cbrt(x) * t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 0.0

   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

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

      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 50.3%

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

   5. Simplified 50.3%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

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

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

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

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

      8. rem-cube-cbrt N/A

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

      9. cube-neg N/A

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

      10. neg-mul-1 N/A

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

      11. unpow-prod-down N/A

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

      12. metadata-eval N/A

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

      13. rem-cube-cbrt N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. rem-cube-cbrt N/A

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

      19. unpow-prod-down N/A

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

      20. neg-mul-1 N/A

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

      21. cube-neg N/A

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

      22. rem-cube-cbrt N/A

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

      23. neg-lowering-neg.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. distribute-neg-frac N/A

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

      2. metadata-eval N/A

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

      3. /-lowering-/.f64 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. cbrt-undiv N/A

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

       2. associate-/l/ N/A

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

       3. metadata-eval N/A

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

       4. distribute-lft-neg-in N/A

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

       5. frac-2neg N/A

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

       6. cbrt-div N/A

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

       7. metadata-eval N/A

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

       8. pow1/3 N/A

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

       9. pow2 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - \color{blue}{\frac{4}{3}}\right)}}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - 4 \cdot \color{blue}{\frac{1}{3}}\right)}}\right)\right) \]

       14. pow-div N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{{x}^{2}}{\color{blue}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}}}\right)\right) \]

       15. pow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\color{blue}{x}}^{\left(4 \cdot \frac{1}{3}\right)}}}\right)\right) \]

       16. pow-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\left({x}^{4}\right)}^{\color{blue}{\frac{1}{3}}}}}\right)\right) \]

       17. pow1/3 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{\sqrt[3]{{x}^{4}}}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}}}{\color{blue}{x \cdot x}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{{x}^{4}}}{x}}{\color{blue}{x}}\right)\right) \]

       20. div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{x}}{x}\right)\right) \]

   11. Applied egg-rr 99.2%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 56.8%

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

      1. rem-exp-log N/A

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

      2. pow1/3 N/A

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

      3. log-pow N/A

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

      4. exp-prod N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({\left(e^{\frac{1}{3}}\right)}^{\color{blue}{\log x}}\right)\right) \]

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

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{1}{3}\right), \log \color{blue}{x}\right)\right) \]

      7. log-lowering-log.f64 56.0%

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

   4. Applied egg-rr 56.0%

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

   6. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{{\left(1 + x\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} - \sqrt[3]{1 + x} \cdot \left(0 - \sqrt[3]{x}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x} \cdot \sqrt[3]{x + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x \cdot \left(x + 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+14)
   (/
    (- (+ x 1.0) x)
    (+
     (pow (+ x 1.0) 0.6666666666666666)
     (+ (pow x 0.6666666666666666) (cbrt (* x (+ x 1.0))))))
   (* 0.3333333333333333 (/ (cbrt x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+14) {
		tmp = ((x + 1.0) - x) / (pow((x + 1.0), 0.6666666666666666) + (pow(x, 0.6666666666666666) + cbrt((x * (x + 1.0)))));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) / x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+14) {
		tmp = ((x + 1.0) - x) / (Math.pow((x + 1.0), 0.6666666666666666) + (Math.pow(x, 0.6666666666666666) + Math.cbrt((x * (x + 1.0)))));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+14)
		tmp = Float64(Float64(Float64(x + 1.0) - x) / Float64((Float64(x + 1.0) ^ 0.6666666666666666) + Float64((x ^ 0.6666666666666666) + cbrt(Float64(x * Float64(x + 1.0))))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+14], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e14

   1. Initial program 56.8%

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

      1. rem-exp-log N/A

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

      2. pow1/3 N/A

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

      3. log-pow N/A

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

      4. exp-prod N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({\left(e^{\frac{1}{3}}\right)}^{\color{blue}{\log x}}\right)\right) \]

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

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

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{1}{3}\right), \log \color{blue}{x}\right)\right) \]

      7. log-lowering-log.f64 56.0%

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

   4. Applied egg-rr 56.0%

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

   6. Applied egg-rr 97.1%

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

   if 4e14 < 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

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

      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 50.3%

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

   5. Simplified 50.3%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

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

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

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

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

      8. rem-cube-cbrt N/A

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

      9. cube-neg N/A

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

      10. neg-mul-1 N/A

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

      11. unpow-prod-down N/A

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

      12. metadata-eval N/A

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

      13. rem-cube-cbrt N/A

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

      14. *-commutative N/A

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

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

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. rem-cube-cbrt N/A

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

      19. unpow-prod-down N/A

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

      20. neg-mul-1 N/A

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

      21. cube-neg N/A

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

      22. rem-cube-cbrt N/A

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

      23. neg-lowering-neg.f64 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. distribute-neg-frac N/A

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

      2. metadata-eval N/A

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

      3. /-lowering-/.f64 98.5%

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

   9. Applied egg-rr 98.5%

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

       1. cbrt-undiv N/A

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

       2. associate-/l/ N/A

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

       3. metadata-eval N/A

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

       4. distribute-lft-neg-in N/A

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

       5. frac-2neg N/A

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

       6. cbrt-div N/A

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

       7. metadata-eval N/A

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

       8. pow1/3 N/A

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

       9. pow2 N/A

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

       10. pow-pow N/A

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

       11. metadata-eval N/A

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

       12. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - \color{blue}{\frac{4}{3}}\right)}}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - 4 \cdot \color{blue}{\frac{1}{3}}\right)}}\right)\right) \]

       14. pow-div N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{{x}^{2}}{\color{blue}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}}}\right)\right) \]

       15. pow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\color{blue}{x}}^{\left(4 \cdot \frac{1}{3}\right)}}}\right)\right) \]

       16. pow-pow N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\left({x}^{4}\right)}^{\color{blue}{\frac{1}{3}}}}}\right)\right) \]

       17. pow1/3 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{\sqrt[3]{{x}^{4}}}}\right)\right) \]

       18. clear-num N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}}}{\color{blue}{x \cdot x}}\right)\right) \]

       19. associate-/r* N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{{x}^{4}}}{x}}{\color{blue}{x}}\right)\right) \]

       20. div-inv N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{x}}{x}\right)\right) \]

   11. Applied egg-rr 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{{\left(x + 1\right)}^{0.6666666666666666} + \left({x}^{0.6666666666666666} + \sqrt[3]{x \cdot \left(x + 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 2.0× 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 6.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.9%

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

5. Simplified 50.9%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

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

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

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

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

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

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

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

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

   8. rem-cube-cbrt N/A

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

   9. cube-neg N/A

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

   10. neg-mul-1 N/A

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

   11. unpow-prod-down N/A

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

   12. metadata-eval N/A

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

   13. rem-cube-cbrt N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

   18. rem-cube-cbrt N/A

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

   19. unpow-prod-down N/A

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

   20. neg-mul-1 N/A

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

   21. cube-neg N/A

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

   22. rem-cube-cbrt N/A

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

   23. neg-lowering-neg.f64 96.8%

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

7. Applied egg-rr 96.8%

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

   1. distribute-neg-frac N/A

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

   2. metadata-eval N/A

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

   3. /-lowering-/.f64 96.8%

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

9. Applied egg-rr 96.8%

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

    1. cbrt-undiv N/A

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

    2. associate-/l/ N/A

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

    3. metadata-eval N/A

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

    4. distribute-lft-neg-in N/A

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

    5. frac-2neg N/A

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

    6. cbrt-div N/A

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

    7. metadata-eval N/A

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

    8. pow1/3 N/A

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

    9. pow2 N/A

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

    10. pow-pow N/A

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

    11. metadata-eval N/A

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

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - \color{blue}{\frac{4}{3}}\right)}}\right)\right) \]

    13. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{{x}^{\left(2 - 4 \cdot \color{blue}{\frac{1}{3}}\right)}}\right)\right) \]

    14. pow-div N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{{x}^{2}}{\color{blue}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}}}\right)\right) \]

    15. pow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\color{blue}{x}}^{\left(4 \cdot \frac{1}{3}\right)}}}\right)\right) \]

    16. pow-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{{\left({x}^{4}\right)}^{\color{blue}{\frac{1}{3}}}}}\right)\right) \]

    17. pow1/3 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{1}{\frac{x \cdot x}{\sqrt[3]{{x}^{4}}}}\right)\right) \]

    18. clear-num N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}}}{\color{blue}{x \cdot x}}\right)\right) \]

    19. associate-/r* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\frac{\sqrt[3]{{x}^{4}}}{x}}{\color{blue}{x}}\right)\right) \]

    20. div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{x}}{x}\right)\right) \]

11. Applied egg-rr 97.5%

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

Alternative 4: 88.9% 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 6.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.9%

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

5. Simplified 50.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 89.1%

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

7. Applied egg-rr 89.1%

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

8. Final simplification 89.1%

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

Alternative 5: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

   1. rem-exp-log N/A

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

   2. pow1/3 N/A

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

   3. log-pow N/A

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

   4. exp-prod N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({\left(e^{\frac{1}{3}}\right)}^{\color{blue}{\log x}}\right)\right) \]

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

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

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{1}{3}\right), \log \color{blue}{x}\right)\right) \]

   7. log-lowering-log.f64 6.9%

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

4. Applied egg-rr 6.9%

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

5. Taylor expanded in x around inf

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

   1. cbrt-lowering-cbrt.f64 5.4%

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

7. Simplified 5.4%

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

Alternative 6: 4.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.7%

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

   1. rem-exp-log N/A

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

   2. pow1/3 N/A

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

   3. log-pow N/A

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

   4. exp-prod N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \left({\left(e^{\frac{1}{3}}\right)}^{\color{blue}{\log x}}\right)\right) \]

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

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

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\frac{1}{3}\right), \log \color{blue}{x}\right)\right) \]

   7. log-lowering-log.f64 6.9%

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

4. Applied egg-rr 6.9%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

      \[\leadsto x \cdot 0 \]

   4. mul0-rgt 4.1%

      \[\leadsto 0 \]

7. Simplified 4.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024192 
(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)))