2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.2%

Time: 13.8s

Alternatives: 17

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{x + 1}\\ \mathbf{if}\;x \leq 155000000:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot x}{{t\_0}^{3}}}{\frac{{\left(x + 1\right)}^{1.3333333333333333}}{{t\_0}^{2}} + \frac{{x}^{0.6666666666666666}}{t\_0} \cdot \frac{{x}^{0.6666666666666666} + {\left(x + 1\right)}^{0.6666666666666666}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt x) (cbrt (+ x 1.0)))))
   (if (<= x 155000000.0)
     (/
      (/ (- (* (+ x 1.0) (+ x 1.0)) (* x x)) (pow t_0 3.0))
      (+
       (/ (pow (+ x 1.0) 1.3333333333333333) (pow t_0 2.0))
       (*
        (/ (pow x 0.6666666666666666) t_0)
        (/
         (+ (pow x 0.6666666666666666) (pow (+ x 1.0) 0.6666666666666666))
         t_0))))
     (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x))))))

C

double code(double x) {
	double t_0 = cbrt(x) + cbrt((x + 1.0));
	double tmp;
	if (x <= 155000000.0) {
		tmp = ((((x + 1.0) * (x + 1.0)) - (x * x)) / pow(t_0, 3.0)) / ((pow((x + 1.0), 1.3333333333333333) / pow(t_0, 2.0)) + ((pow(x, 0.6666666666666666) / t_0) * ((pow(x, 0.6666666666666666) + pow((x + 1.0), 0.6666666666666666)) / t_0)));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) + Math.cbrt((x + 1.0));
	double tmp;
	if (x <= 155000000.0) {
		tmp = ((((x + 1.0) * (x + 1.0)) - (x * x)) / Math.pow(t_0, 3.0)) / ((Math.pow((x + 1.0), 1.3333333333333333) / Math.pow(t_0, 2.0)) + ((Math.pow(x, 0.6666666666666666) / t_0) * ((Math.pow(x, 0.6666666666666666) + Math.pow((x + 1.0), 0.6666666666666666)) / t_0)));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) + cbrt(Float64(x + 1.0)))
	tmp = 0.0
	if (x <= 155000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x + 1.0) * Float64(x + 1.0)) - Float64(x * x)) / (t_0 ^ 3.0)) / Float64(Float64((Float64(x + 1.0) ^ 1.3333333333333333) / (t_0 ^ 2.0)) + Float64(Float64((x ^ 0.6666666666666666) / t_0) * Float64(Float64((x ^ 0.6666666666666666) + (Float64(x + 1.0) ^ 0.6666666666666666)) / t_0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 155000000.0], N[(N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 1.3333333333333333], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[x, 0.6666666666666666], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55e8

   1. Initial program 80.1%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 79.8%

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

   4. Applied egg-rr 79.8%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]

   6. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot x}{{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}^{3}}}{\frac{{\left(x + 1\right)}^{1.3333333333333333}}{{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}^{2}} + \frac{{x}^{0.6666666666666666}}{\sqrt[3]{x} + \sqrt[3]{x + 1}} \cdot \frac{{\left(x + 1\right)}^{0.6666666666666666} - \left(0 - {x}^{0.6666666666666666}\right)}{\sqrt[3]{x} + \sqrt[3]{x + 1}}}} \]

   if 1.55e8 < x

   1. Initial program 5.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.2%

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

   5. Simplified 52.2%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.8%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 155000000:\\ \;\;\;\;\frac{\frac{\left(x + 1\right) \cdot \left(x + 1\right) - x \cdot x}{{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}^{3}}}{\frac{{\left(x + 1\right)}^{1.3333333333333333}}{{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}^{2}} + \frac{{x}^{0.6666666666666666}}{\sqrt[3]{x} + \sqrt[3]{x + 1}} \cdot \frac{{x}^{0.6666666666666666} + {\left(x + 1\right)}^{0.6666666666666666}}{\sqrt[3]{x} + \sqrt[3]{x + 1}}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 125000000:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) \cdot \left({x}^{0.6666666666666666} + {\left(x + 1\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 125000000.0)
   (/
    (- (pow (+ x 1.0) 1.3333333333333333) (pow x 1.3333333333333333))
    (*
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (+ (pow x 0.6666666666666666) (pow (+ x 1.0) 0.6666666666666666))))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 125000000.0) {
		tmp = (pow((x + 1.0), 1.3333333333333333) - pow(x, 1.3333333333333333)) / ((cbrt(x) + cbrt((x + 1.0))) * (pow(x, 0.6666666666666666) + pow((x + 1.0), 0.6666666666666666)));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 125000000.0) {
		tmp = (Math.pow((x + 1.0), 1.3333333333333333) - Math.pow(x, 1.3333333333333333)) / ((Math.cbrt(x) + Math.cbrt((x + 1.0))) * (Math.pow(x, 0.6666666666666666) + Math.pow((x + 1.0), 0.6666666666666666)));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 125000000.0)
		tmp = Float64(Float64((Float64(x + 1.0) ^ 1.3333333333333333) - (x ^ 1.3333333333333333)) / Float64(Float64(cbrt(x) + cbrt(Float64(x + 1.0))) * Float64((x ^ 0.6666666666666666) + (Float64(x + 1.0) ^ 0.6666666666666666))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 125000000.0], N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 1.3333333333333333], $MachinePrecision] - N[Power[x, 1.3333333333333333], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25e8

   1. Initial program 85.2%

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

      1. flip-- N/A

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

      2. flip-- N/A

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

      3. associate-/l/ N/A

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

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

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

   4. Applied egg-rr 90.1%

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

   if 1.25e8 < x

   1. Initial program 5.6%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.5%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 125000000:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right) \cdot \left({x}^{0.6666666666666666} + {\left(x + 1\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\frac{-1}{\frac{\sqrt[3]{x} + \sqrt[3]{x + 1}}{{x}^{0.6666666666666666} - {\left(x + 1\right)}^{0.6666666666666666}}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 58000000.0)
   (/
    -1.0
    (/
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (- (pow x 0.6666666666666666) (pow (+ x 1.0) 0.6666666666666666))))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = -1.0 / ((cbrt(x) + cbrt((x + 1.0))) / (pow(x, 0.6666666666666666) - pow((x + 1.0), 0.6666666666666666)));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = -1.0 / ((Math.cbrt(x) + Math.cbrt((x + 1.0))) / (Math.pow(x, 0.6666666666666666) - Math.pow((x + 1.0), 0.6666666666666666)));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 58000000.0)
		tmp = Float64(-1.0 / Float64(Float64(cbrt(x) + cbrt(Float64(x + 1.0))) / Float64((x ^ 0.6666666666666666) - (Float64(x + 1.0) ^ 0.6666666666666666))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 58000000.0], N[(-1.0 / N[(N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 0.6666666666666666], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e7

   1. Initial program 88.7%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 89.4%

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

   4. Applied egg-rr 89.4%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow1/3 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

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

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

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

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

      6. +-commutative N/A

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

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

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

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

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

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

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

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

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

      11. pow2 N/A

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

      12. pow1/3 N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. pow2 N/A

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

      16. pow1/3 N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

   6. Applied egg-rr 91.8%

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

   if 5.8e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\frac{-1}{\frac{\sqrt[3]{x} + \sqrt[3]{x + 1}}{{x}^{0.6666666666666666} - {\left(x + 1\right)}^{0.6666666666666666}}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\left({\left(x + 1\right)}^{0.6666666666666666} - {x}^{0.6666666666666666}\right) \cdot \frac{1}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 58000000.0)
   (*
    (- (pow (+ x 1.0) 0.6666666666666666) (pow x 0.6666666666666666))
    (/ 1.0 (+ (cbrt x) (cbrt (+ x 1.0)))))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = (pow((x + 1.0), 0.6666666666666666) - pow(x, 0.6666666666666666)) * (1.0 / (cbrt(x) + cbrt((x + 1.0))));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = (Math.pow((x + 1.0), 0.6666666666666666) - Math.pow(x, 0.6666666666666666)) * (1.0 / (Math.cbrt(x) + Math.cbrt((x + 1.0))));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 58000000.0)
		tmp = Float64(Float64((Float64(x + 1.0) ^ 0.6666666666666666) - (x ^ 0.6666666666666666)) * Float64(1.0 / Float64(cbrt(x) + cbrt(Float64(x + 1.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 58000000.0], N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] - N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e7

   1. Initial program 88.7%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. pow1/3 N/A

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

      12. pow1/3 N/A

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

      13. pow-prod-up N/A

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

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

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

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

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

      16. metadata-eval N/A

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

      17. pow1/3 N/A

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

      18. pow1/3 N/A

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

   4. Applied egg-rr 91.8%

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

   if 5.8e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\left({\left(x + 1\right)}^{0.6666666666666666} - {x}^{0.6666666666666666}\right) \cdot \frac{1}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{0.6666666666666666} - {x}^{0.6666666666666666}}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 58000000.0)
   (/
    (- (pow (+ x 1.0) 0.6666666666666666) (pow x 0.6666666666666666))
    (+ (cbrt x) (cbrt (+ x 1.0))))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = (pow((x + 1.0), 0.6666666666666666) - pow(x, 0.6666666666666666)) / (cbrt(x) + cbrt((x + 1.0)));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 58000000.0) {
		tmp = (Math.pow((x + 1.0), 0.6666666666666666) - Math.pow(x, 0.6666666666666666)) / (Math.cbrt(x) + Math.cbrt((x + 1.0)));
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 58000000.0)
		tmp = Float64(Float64((Float64(x + 1.0) ^ 0.6666666666666666) - (x ^ 0.6666666666666666)) / Float64(cbrt(x) + cbrt(Float64(x + 1.0))));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 58000000.0], N[(N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision] - N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e7

   1. Initial program 88.7%

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

      1. flip-- N/A

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

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

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

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

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. pow-prod-up N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. pow1/3 N/A

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

      11. pow1/3 N/A

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

      12. pow-prod-up N/A

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

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

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

      14. metadata-eval N/A

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

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

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

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

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

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

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

      18. cbrt-lowering-cbrt.f64 91.8%

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

   4. Applied egg-rr 91.8%

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

   if 5.8e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 58000000:\\ \;\;\;\;\frac{{\left(x + 1\right)}^{0.6666666666666666} - {x}^{0.6666666666666666}}{\sqrt[3]{x} + \sqrt[3]{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.25}, 0 - {x}^{0.08333333333333333}, {\left(x + 1\right)}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 23000000.0)
   (fma
    (pow x 0.25)
    (- 0.0 (pow x 0.08333333333333333))
    (pow (+ x 1.0) 0.3333333333333333))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 23000000.0) {
		tmp = fma(pow(x, 0.25), (0.0 - pow(x, 0.08333333333333333)), pow((x + 1.0), 0.3333333333333333));
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 23000000.0)
		tmp = fma((x ^ 0.25), Float64(0.0 - (x ^ 0.08333333333333333)), (Float64(x + 1.0) ^ 0.3333333333333333));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 23000000.0], N[(N[Power[x, 0.25], $MachinePrecision] * N[(0.0 - N[Power[x, 0.08333333333333333], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.3e7

   1. Initial program 88.7%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 89.4%

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

   4. Applied egg-rr 89.4%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow1/3 N/A

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

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

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

      3. +-lowering-+.f64 90.5%

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

   6. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {x}^{0.3333333333333333} \]
   7. Step-by-step derivation

      1. pow1/3 N/A

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

      2. pow1/3 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left({x}^{\left(\frac{1}{4} + \frac{1}{12}\right)}\right)\right) + \sqrt[3]{x + 1} \]

      7. pow-prod-up N/A

         \[\leadsto \left(\mathsf{neg}\left({x}^{\frac{1}{4}} \cdot {x}^{\frac{1}{12}}\right)\right) + \sqrt[3]{\color{blue}{x} + 1} \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto {x}^{\frac{1}{4}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{12}}\right)\right) + \sqrt[3]{\color{blue}{x + 1}} \]

      9. fma-define N/A

         \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{4}}, \color{blue}{\mathsf{neg}\left({x}^{\frac{1}{12}}\right)}, \sqrt[3]{x + 1}\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\left({x}^{\frac{1}{4}}\right), \color{blue}{\left(\mathsf{neg}\left({x}^{\frac{1}{12}}\right)\right)}, \left(\sqrt[3]{x + 1}\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \left(\mathsf{neg}\left(\color{blue}{{x}^{\frac{1}{12}}}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{neg.f64}\left(\left({x}^{\frac{1}{12}}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{12}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

      14. pow1/3 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{12}\right)\right), \left({\left(x + 1\right)}^{\frac{1}{3}}\right)\right) \]

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

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{12}\right)\right), \mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{3}\right)\right) \]

      16. +-lowering-+.f64 90.5%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{4}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{12}\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{3}\right)\right) \]

   8. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, -{x}^{0.08333333333333333}, {\left(x + 1\right)}^{0.3333333333333333}\right)} \]

   if 2.3e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 23000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.25}, 0 - {x}^{0.08333333333333333}, {\left(x + 1\right)}^{0.3333333333333333}\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 40000000.0)
   (- (pow (+ x 1.0) 0.3333333333333333) (pow x 0.3333333333333333))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 40000000.0) {
		tmp = pow((x + 1.0), 0.3333333333333333) - pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 40000000.0) {
		tmp = Math.pow((x + 1.0), 0.3333333333333333) - Math.pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 40000000.0)
		tmp = Float64((Float64(x + 1.0) ^ 0.3333333333333333) - (x ^ 0.3333333333333333));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 40000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e7

   1. Initial program 88.7%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 89.4%

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

   4. Applied egg-rr 89.4%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]
   5. Step-by-step derivation

      1. pow1/3 N/A

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

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

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

      3. +-lowering-+.f64 90.5%

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

   6. Applied egg-rr 90.5%

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

   if 4e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40000000:\\ \;\;\;\;{\left(x + 1\right)}^{0.3333333333333333} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 22000000:\\ \;\;\;\;\sqrt[3]{x + 1} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 22000000.0)
   (- (cbrt (+ x 1.0)) (pow x 0.3333333333333333))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 22000000.0) {
		tmp = cbrt((x + 1.0)) - pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 22000000.0) {
		tmp = Math.cbrt((x + 1.0)) - Math.pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 22000000.0)
		tmp = Float64(cbrt(Float64(x + 1.0)) - (x ^ 0.3333333333333333));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 22000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e7

   1. Initial program 88.7%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 89.4%

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

   4. Applied egg-rr 89.4%

      \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{x}^{0.3333333333333333}} \]

   if 2.2e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 22000000:\\ \;\;\;\;\sqrt[3]{x + 1} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 46000000.0)
   (- (cbrt (+ x 1.0)) (cbrt x))
   (* 0.3333333333333333 (* (cbrt x) (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = cbrt((x + 1.0)) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = Math.cbrt((x + 1.0)) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * (Math.cbrt(x) * (1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 46000000.0)
		tmp = Float64(cbrt(Float64(x + 1.0)) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * Float64(cbrt(x) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 46000000.0], N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e7

   1. Initial program 88.7%

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

   if 4.6e7 < x

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 52.3%

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

   5. Simplified 52.3%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. inv-pow N/A

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

      4. cbrt-prod N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

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

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

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

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

      9. metadata-eval 97.4%

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

   7. Applied egg-rr 97.4%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. cbrt-unprod N/A

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

      5. pow1/3 N/A

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

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

      7. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      8. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

      9. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

      10. sqr-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      11. swap-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

      12. unpow-prod-down N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

      15. distribute-rgt-neg-out N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      16. pow-prod-up N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      18. pow1/3 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      19. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

      20. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

   9. Applied egg-rr 97.3%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

   11. Applied egg-rr 98.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\sqrt[3]{x} \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+155}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6e+155)
   (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x)))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6e+155) {
		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 <= 1.6e+155) {
		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 <= 1.6e+155)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) / x)));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6e+155], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000006e155

   1. Initial program 11.6%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.0%

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

   5. Simplified 93.0%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 93.0%

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

   7. Applied egg-rr 93.0%

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

   if 1.60000000000000006e155 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.8%

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

   5. Simplified 4.8%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow2 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-lowering-pow.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

Alternative 11: 91.8% accurate, 1.8× 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 11.6%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.0%

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

   5. Simplified 93.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.8%

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

   5. Simplified 4.8%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow2 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. pow-lowering-pow.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

Alternative 12: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 51.6%

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

5. Simplified 51.6%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 95.3%

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

7. Applied egg-rr 95.3%

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

   1. metadata-eval N/A

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

   2. pow-pow N/A

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

   3. pow2 N/A

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

   4. cbrt-unprod N/A

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

   5. pow1/3 N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)}^{-1}\right)\right) \]

   7. pow-prod-up N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\left(x \cdot x\right)}^{\frac{1}{6}} \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

   8. pow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot {\left(x \cdot x\right)}^{\frac{1}{6}}\right)}^{-1}\right)\right) \]

   9. pow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right)}^{-1}\right)\right) \]

   10. sqr-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right) \cdot \left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right) \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

   11. swap-sqr N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right) \cdot \left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)\right)}^{-1}\right)\right) \]

   12. unpow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1} \cdot \color{blue}{{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right), \color{blue}{\left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), -1\right), \left({\color{blue}{\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}}^{-1}\right)\right)\right) \]

   15. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{6}} \cdot {x}^{\frac{1}{6}}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

   16. pow-prod-up N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left({x}^{\frac{1}{3}}\right)\right), -1\right), \left({\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

   18. pow1/3 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\sqrt[3]{x}\right)\right), -1\right), \left({\left({\color{blue}{x}}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

   19. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \sqrt[3]{x}\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

   20. --lowering--.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\sqrt[3]{x}\right)\right), -1\right), \left({\left(\color{blue}{{x}^{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \left({\left({x}^{\color{blue}{\frac{1}{6}}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right)}^{-1}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{cbrt.f64}\left(x\right)\right), -1\right), \mathsf{pow.f64}\left(\left({x}^{\frac{1}{6}} \cdot \left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)\right), \color{blue}{-1}\right)\right)\right) \]

9. Applied egg-rr 95.2%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left({\left(0 - \sqrt[3]{x}\right)}^{-1} \cdot {\left(0 - \sqrt[3]{x}\right)}^{-1}\right)} \]

11. Applied egg-rr 96.0%

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

12. Final simplification 96.0%

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

Alternative 13: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{1}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (/ 1.0 (pow x 0.6666666666666666))))

C

double code(double x) {
	return 0.3333333333333333 * (1.0 / pow(x, 0.6666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (1.0d0 / (x ** 0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * (1.0 / Math.pow(x, 0.6666666666666666));
}

Python

def code(x):
	return 0.3333333333333333 * (1.0 / math.pow(x, 0.6666666666666666))

Julia

function code(x)
	return Float64(0.3333333333333333 * Float64(1.0 / (x ^ 0.6666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (1.0 / (x ^ 0.6666666666666666));
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 51.6%

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

5. Simplified 51.6%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

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

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

   5. pow2 N/A

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

   6. pow1/3 N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

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

   9. pow-lowering-pow.f64 87.9%

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

7. Applied egg-rr 87.9%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 14: 88.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 51.6%

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

5. Simplified 51.6%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

   4. un-div-inv N/A

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

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

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

   6. pow2 N/A

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

   7. pow1/3 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. pow-lowering-pow.f64 87.9%

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

7. Applied egg-rr 87.9%

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

Alternative 15: 88.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 51.6%

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

5. Simplified 51.6%

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

   1. *-commutative N/A

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

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

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

   3. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 87.9%

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

7. Applied egg-rr 87.9%

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

8. Final simplification 87.9%

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

Alternative 16: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 8.4%

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

3. Taylor expanded in x around 0

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

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

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

   2. cbrt-lowering-cbrt.f64 1.8%

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

5. Simplified 1.8%

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

   1. sub-neg N/A

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

   2. sub0-neg N/A

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

   3. +-commutative N/A

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

7. Applied egg-rr 5.5%

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

8. Taylor expanded in x around inf

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

   1. cbrt-lowering-cbrt.f64 5.5%

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

10. Simplified 5.5%

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

Alternative 17: 4.2% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 8.4%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

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

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

   6. metadata-eval 9.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(x, 1\right)\right), \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), 2\right)\right) \]

4. Applied egg-rr 9.4%

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

   1. pow-pow N/A

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

   2. metadata-eval N/A

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

   3. pow1/3 N/A

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

   4. sub-neg N/A

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

   5. sub0-neg N/A

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

   6. +-commutative N/A

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

   7. sub0-neg N/A

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

   8. pow1/3 N/A

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

   9. metadata-eval N/A

      \[\leadsto \left(\mathsf{neg}\left({x}^{\left(\frac{1}{6} + \frac{1}{6}\right)}\right)\right) + \sqrt[3]{x + 1} \]

   10. pow-prod-up N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. fma-define N/A

       \[\leadsto \mathsf{fma}\left({x}^{\frac{1}{6}}, \color{blue}{\mathsf{neg}\left({x}^{\frac{1}{6}}\right)}, \sqrt[3]{x + 1}\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\left({x}^{\frac{1}{6}}\right), \color{blue}{\left(\mathsf{neg}\left({x}^{\frac{1}{6}}\right)\right)}, \left(\sqrt[3]{x + 1}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \left(\mathsf{neg}\left(\color{blue}{{x}^{\frac{1}{6}}}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \mathsf{neg.f64}\left(\left({x}^{\frac{1}{6}}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left(\sqrt[3]{x + 1}\right)\right) \]

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

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{cbrt.f64}\left(\left(x + 1\right)\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{cbrt.f64}\left(\left(1 + x\right)\right)\right) \]

   19. +-lowering-+.f64 9.4%

       \[\leadsto \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right), \mathsf{neg.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{cbrt.f64}\left(\mathsf{+.f64}\left(1, x\right)\right)\right) \]

6. Applied egg-rr 9.4%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

      \[\leadsto x \cdot 0 \]

   4. *-lowering-*.f64 4.1%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{0}\right) \]

9. Simplified 4.1%

   \[\leadsto \color{blue}{x \cdot 0} \]
10. Step-by-step derivation

    1. mul0-rgt 4.1%

       \[\leadsto 0 \]

11. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{0} \]
12. 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 2024154 
(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)))