2cbrt (problem 3.3.4)

Percentage Accurate: 7.3% → 98.8%

Time: 11.6s

Alternatives: 16

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{x}}\\ \mathbf{if}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)} + \left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(t\_0 + t\_0\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 x))))
   (if (<= x 3e+14)
     (/
      (+ x (- 1.0 x))
      (+
       (cbrt (* (+ 1.0 x) (+ 1.0 x)))
       (+ (pow x 0.6666666666666666) (cbrt (fma x x x)))))
     (/ 1.0 (* x (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x)))) (+ t_0 t_0)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 / x));
	double tmp;
	if (x <= 3e+14) {
		tmp = (x + (1.0 - x)) / (cbrt(((1.0 + x) * (1.0 + x))) + (pow(x, 0.6666666666666666) + cbrt(fma(x, x, x))));
	} else {
		tmp = 1.0 / (x * (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (t_0 + t_0)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 / x))
	tmp = 0.0
	if (x <= 3e+14)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(cbrt(Float64(Float64(1.0 + x) * Float64(1.0 + x))) + Float64((x ^ 0.6666666666666666) + cbrt(fma(x, x, x)))));
	else
		tmp = Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(t_0 + t_0))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 3e+14], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3e14

   1. Initial program 69.9%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 67.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 97.7%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. unpow1/3 N/A

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

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

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 98.6

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

   8. Applied egg-rr 98.6%

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

   if 3e14 < x

   1. Initial program 4.2%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 5.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 5.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 4.5%

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

   7. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

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

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

      10. unpow2 N/A

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

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

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

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

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

   9. Simplified 98.8%

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

   10. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 98.8

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

   12. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)} + \left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x}}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 51.3

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. associate-/r* N/A

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

      3. div-inv N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. pow1/3 N/A

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

      12. cbrt-div N/A

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

      13. metadata-eval N/A

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

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

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

      15. cbrt-lowering-cbrt.f64 98.4

          \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \frac{1}{\color{blue}{\sqrt[3]{x}}}\right) \]

   7. Applied egg-rr 98.4%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

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

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

      4. /-lowering-/.f64 98.5

         \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\frac{1}{x}}}\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right) \]

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

   1. Initial program 69.9%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 67.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 97.7%

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

      1. metadata-eval N/A

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

      2. pow-sqr N/A

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

      3. pow-prod-down N/A

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

      4. pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. unpow1/3 N/A

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

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

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

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

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

      13. +-commutative N/A

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

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

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 98.6

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

   8. Applied egg-rr 98.6%

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

4. Final simplification 98.5%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 51.3

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. associate-/r* N/A

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

      3. div-inv N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. pow1/3 N/A

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

      12. cbrt-div N/A

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

      13. metadata-eval N/A

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

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

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

      15. cbrt-lowering-cbrt.f64 98.4

          \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \frac{1}{\color{blue}{\sqrt[3]{x}}}\right) \]

   7. Applied egg-rr 98.4%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

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

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

      4. /-lowering-/.f64 98.5

         \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\frac{1}{x}}}\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right) \]

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

   1. Initial program 69.9%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 67.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 97.7%

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

      1. rem-cbrt-cube N/A

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

      2. pow-pow N/A

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

      3. metadata-eval N/A

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

      4. pow2 N/A

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

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

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

      6. *-lowering-*.f64 98.4

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

   8. Applied egg-rr 98.4%

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

4. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 51.3

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. associate-/r* N/A

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

      3. div-inv N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. pow1/3 N/A

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

      12. cbrt-div N/A

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

      13. metadata-eval N/A

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

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

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

      15. cbrt-lowering-cbrt.f64 98.4

          \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \frac{1}{\color{blue}{\sqrt[3]{x}}}\right) \]

   7. Applied egg-rr 98.4%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

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

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

      4. /-lowering-/.f64 98.5

         \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\frac{1}{x}}}\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right) \]

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

   1. Initial program 69.9%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 67.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 97.7%

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

      9. accelerator-lowering-fma.f64 97.9

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 98.4%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      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 51.3

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

   5. Simplified 51.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. associate-/r* N/A

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

      3. div-inv N/A

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

      4. unpow-prod-down N/A

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

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

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

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

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

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

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

      11. pow1/3 N/A

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

      12. cbrt-div N/A

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

      13. metadata-eval N/A

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

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

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

      15. cbrt-lowering-cbrt.f64 98.4

          \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \frac{1}{\color{blue}{\sqrt[3]{x}}}\right) \]

   7. Applied egg-rr 98.4%

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

      1. metadata-eval N/A

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

      2. cbrt-div N/A

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

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

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

      4. /-lowering-/.f64 98.5

         \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\frac{1}{x}}}\right) \]

   9. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right) \]

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

   1. Initial program 69.9%

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

      1. pow1/3 N/A

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

      2. sqr-pow N/A

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

      3. pow2 N/A

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

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

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

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

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

      6. metadata-eval 67.6

         \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
   5. Step-by-step derivation

      1. pow-pow N/A

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

      2. metadata-eval N/A

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

      3. pow1/3 N/A

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

      4. flip3-- N/A

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

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

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

      6. rem-cube-cbrt N/A

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

      7. rem-cube-cbrt N/A

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

      8. associate--l+ N/A

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

      9. sub-neg N/A

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

      10. *-rgt-identity N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. --lowering--.f64 N/A

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

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

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

   6. Applied egg-rr 97.7%

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

      1. associate-+r- N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

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

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

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

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

      14. accelerator-lowering-fma.f64 97.7

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

   8. Applied egg-rr 97.7%

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

4. Final simplification 98.4%

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

Alternative 6: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

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

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

   6. metadata-eval 8.6

      \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

4. Applied egg-rr 8.6%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

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

   3. pow1/3 N/A

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

   4. flip3-- N/A

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

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

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

   6. rem-cube-cbrt N/A

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

   7. rem-cube-cbrt N/A

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

   8. associate--l+ N/A

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

   9. sub-neg N/A

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

   10. *-rgt-identity N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

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

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. --lowering--.f64 N/A

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

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

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

6. Applied egg-rr 8.9%

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

7. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

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

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

9. Simplified 97.4%

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

    1. cbrt-div N/A

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

    2. metadata-eval N/A

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

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

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

    4. cbrt-lowering-cbrt.f64 97.5

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

11. Applied egg-rr 97.5%

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

12. Final simplification 97.5%

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

Alternative 7: 97.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

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

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

   6. metadata-eval 8.6

      \[\leadsto \sqrt[3]{x + 1} - {\left({x}^{\color{blue}{0.16666666666666666}}\right)}^{2} \]

4. Applied egg-rr 8.6%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left({x}^{0.16666666666666666}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

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

   3. pow1/3 N/A

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

   4. flip3-- N/A

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

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

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

   6. rem-cube-cbrt N/A

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

   7. rem-cube-cbrt N/A

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

   8. associate--l+ N/A

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

   9. sub-neg N/A

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

   10. *-rgt-identity N/A

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

   11. cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

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

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. --lowering--.f64 N/A

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

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

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

6. Applied egg-rr 8.9%

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

7. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

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

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

   10. unpow2 N/A

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

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

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

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

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

9. Simplified 97.4%

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

10. Final simplification 97.4%

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

Alternative 8: 96.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 51.2

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

5. Simplified 51.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. associate-/r* N/A

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

   3. div-inv N/A

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

   4. unpow-prod-down N/A

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

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

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. pow1/3 N/A

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

   12. cbrt-div N/A

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

   13. metadata-eval N/A

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

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

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

   15. cbrt-lowering-cbrt.f64 96.1

       \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \frac{1}{\color{blue}{\sqrt[3]{x}}}\right) \]

7. Applied egg-rr 96.1%

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

   1. metadata-eval N/A

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

   2. cbrt-div N/A

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

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

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

   4. /-lowering-/.f64 96.2

      \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \sqrt[3]{\color{blue}{\frac{1}{x}}}\right) \]

9. Applied egg-rr 96.2%

   \[\leadsto 0.3333333333333333 \cdot \left(\frac{1}{\sqrt[3]{x}} \cdot \color{blue}{\sqrt[3]{\frac{1}{x}}}\right) \]
10. Add Preprocessing

Alternative 9: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (/ 1.0 (pow (cbrt x) 2.0))))

C

double code(double x) {
	return 0.3333333333333333 * (1.0 / pow(cbrt(x), 2.0));
}

Java

public static double code(double x) {
	return 0.3333333333333333 * (1.0 / Math.pow(Math.cbrt(x), 2.0));
}

Julia

function code(x)
	return Float64(0.3333333333333333 * Float64(1.0 / (cbrt(x) ^ 2.0)))
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 51.2

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

5. Simplified 51.2%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

   8. pow1/3 N/A

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

   9. cbrt-div N/A

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

   10. metadata-eval N/A

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

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

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

   12. cbrt-lowering-cbrt.f64 96.1

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

7. Applied egg-rr 96.1%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. pow2 N/A

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

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

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

   5. cbrt-lowering-cbrt.f64 96.1

      \[\leadsto 0.3333333333333333 \cdot \frac{1}{{\color{blue}{\left(\sqrt[3]{x}\right)}}^{2}} \]

9. Applied egg-rr 96.1%

   \[\leadsto 0.3333333333333333 \cdot \frac{1}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
10. Add Preprocessing

Alternative 10: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.3333333333333333 (pow (cbrt x) -2.0)))

C

double code(double x) {
	return 0.3333333333333333 * pow(cbrt(x), -2.0);
}

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
}

Julia

function code(x)
	return Float64(0.3333333333333333 * (cbrt(x) ^ -2.0))
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 51.2

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

5. Simplified 51.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 96.1

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

7. Applied egg-rr 96.1%

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

Alternative 11: 91.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.3

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

   5. Simplified 94.3%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. pow1/3 N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. pow1/3 N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

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

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

      12. cbrt-lowering-cbrt.f64 93.8

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

   7. Applied egg-rr 93.8%

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

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. cbrt-prod N/A

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

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

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

      5. *-lowering-*.f64 94.5

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

   9. Applied egg-rr 94.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      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 4.7

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

   5. Simplified 4.7%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. pow1/3 N/A

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

      9. pow1/3 N/A

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

      10. pow-prod-up N/A

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

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

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

      12. metadata-eval 89.1

          \[\leadsto \frac{1}{\frac{{x}^{\color{blue}{0.6666666666666666}}}{0.3333333333333333}} \]

   7. Applied egg-rr 89.1%

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

Alternative 12: 91.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.3

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

   5. Simplified 94.3%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. pow1/3 N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. pow1/3 N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

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

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

      12. cbrt-lowering-cbrt.f64 93.8

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

   7. Applied egg-rr 93.8%

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

      1. associate-/r/ N/A

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

      2. /-rgt-identity N/A

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

      3. cbrt-prod N/A

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

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

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

      5. *-lowering-*.f64 94.5

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

   9. Applied egg-rr 94.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      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 4.7

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

   5. Simplified 4.7%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

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

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

      10. metadata-eval 89.1

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

   7. Applied egg-rr 89.1%

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

Alternative 13: 91.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.3

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

   5. Simplified 94.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      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 4.7

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

   5. Simplified 4.7%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

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

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

      10. metadata-eval 89.1

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

   7. Applied egg-rr 89.1%

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

Alternative 14: 88.6% 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 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 51.2

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

5. Simplified 51.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. pow-pow N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 88.5

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

7. Applied egg-rr 88.5%

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

8. Final simplification 88.5%

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

Alternative 15: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 7.2%

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

   1. rem-exp-log N/A

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

   2. pow1/3 N/A

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

   3. log-pow N/A

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

   4. exp-prod N/A

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

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

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

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

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

   7. log-lowering-log.f64 7.4

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

4. Applied egg-rr 7.4%

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

5. Taylor expanded in x around inf

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

   1. cbrt-lowering-cbrt.f64 5.5

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

7. Simplified 5.5%

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

Alternative 16: 4.2% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.2%

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

   1. rem-exp-log N/A

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

   2. pow1/3 N/A

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

   3. log-pow N/A

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

   4. exp-prod N/A

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

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

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

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

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

   7. log-lowering-log.f64 7.4

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

4. Applied egg-rr 7.4%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

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

   4. mul0-rgt 4.1

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

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 2024201 
(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)))