2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.8%

Time: 9.9s

Alternatives: 18

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{1}{\sqrt{x}} \cdot \sqrt[3]{-1}}{\sqrt[3]{-\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\mathsf{fma}\left(x + 1, x + 1, {t\_0}^{3}\right)} \cdot \mathsf{fma}\left(t\_0, t\_0 - {\left(x + 1\right)}^{0.6666666666666666}, {\left(x + 1\right)}^{1.3333333333333333}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (pow x 0.6666666666666666) (cbrt (fma x x x)))))
   (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 0.0)
     (*
      0.3333333333333333
      (/ (* (/ 1.0 (sqrt x)) (cbrt -1.0)) (cbrt (- (sqrt x)))))
     (*
      (/ (+ x (- 1.0 x)) (fma (+ x 1.0) (+ x 1.0) (pow t_0 3.0)))
      (fma
       t_0
       (- t_0 (pow (+ x 1.0) 0.6666666666666666))
       (pow (+ x 1.0) 1.3333333333333333))))))

C

double code(double x) {
	double t_0 = pow(x, 0.6666666666666666) + cbrt(fma(x, x, x));
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * (((1.0 / sqrt(x)) * cbrt(-1.0)) / cbrt(-sqrt(x)));
	} else {
		tmp = ((x + (1.0 - x)) / fma((x + 1.0), (x + 1.0), pow(t_0, 3.0))) * fma(t_0, (t_0 - pow((x + 1.0), 0.6666666666666666)), pow((x + 1.0), 1.3333333333333333));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(0.3333333333333333 * N[(N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Power[-1.0, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[(-N[Sqrt[x], $MachinePrecision]), 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(t$95$0 - N[Power[N[(x + 1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 48.0

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

   5. Simplified 48.0%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.6%

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

   11. Applied egg-rr 98.9%

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

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

   1. Initial program 63.1%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 60.2

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

   4. Applied egg-rr 60.2%

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

   6. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\mathsf{fma}\left(x + 1, x + 1, {\left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}^{3}\right)} \cdot \mathsf{fma}\left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}, \left({x}^{0.6666666666666666} + \sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right) - {\left(x + 1\right)}^{0.6666666666666666}, {\left(x + 1\right)}^{1.3333333333333333}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 48.0

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

   5. Simplified 48.0%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.6%

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

   11. Applied egg-rr 98.9%

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

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

   1. Initial program 63.1%

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

      1. pow1/3 N/A

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

      2. pow-lowering-pow.f64 60.2

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

   4. Applied egg-rr 60.2%

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

      1. flip3-- N/A

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

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

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

      3. rem-cube-cbrt N/A

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

      4. unpow1/3 N/A

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

      5. rem-cube-cbrt N/A

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

      6. associate--l+ N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-rgt-identity N/A

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

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

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

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

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

   6. Applied egg-rr 97.8%

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

4. Final simplification 98.8%

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

Alternative 3: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{\frac{1}{\sqrt{x}} \cdot \sqrt[3]{-1}}{\sqrt[3]{-\sqrt{x}}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.5

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

5. Simplified 48.5%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

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

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

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

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. neg-lowering-neg.f64 96.4

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

7. Applied egg-rr 96.4%

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

   1. rem-square-sqrt N/A

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

   2. associate-/r* N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. inv-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. pow2 N/A

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

   10. sqr-neg N/A

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

   11. rem-square-sqrt N/A

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

   12. pow-pow N/A

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

   13. pow1/2 N/A

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

   14. inv-pow N/A

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

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

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

9. Applied egg-rr 96.4%

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

11. Applied egg-rr 96.7%

    \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}} \cdot \sqrt[3]{-1}}{\sqrt[3]{-\sqrt{x}}}} \]
12. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.5

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

5. Simplified 48.5%

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

   1. associate-/r* N/A

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

   2. frac-2neg N/A

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

   3. cbrt-div N/A

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

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

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

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

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. neg-lowering-neg.f64 96.4

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

7. Applied egg-rr 96.4%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}}} \]
8. Add Preprocessing

Alternative 5: 96.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.5

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

5. Simplified 48.5%

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

   1. associate-/r* N/A

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

   2. cbrt-div N/A

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

   3. pow1/3 N/A

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

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

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

   5. pow1/3 N/A

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

   6. cbrt-div N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. cbrt-lowering-cbrt.f64 96.3

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

7. Applied egg-rr 96.3%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{x}}} \]
8. Add Preprocessing

Alternative 6: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

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

3. Taylor expanded in x around inf

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

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

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

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 48.5

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

5. Simplified 48.5%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 96.3

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

7. Applied egg-rr 96.3%

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

Alternative 7: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\sqrt{x}}}\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{+248}:\\ \;\;\;\;0.3333333333333333 \cdot \left(t\_0 \cdot \sqrt[3]{t\_0 \cdot \frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left(0.3333333333333333 \cdot {x}^{-0.16666666666666666}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (sqrt x)))))
   (if (<= x 1.65e+248)
     (* 0.3333333333333333 (* t_0 (cbrt (* t_0 (/ 1.0 x)))))
     (*
      (/ 1.0 (sqrt x))
      (* 0.3333333333333333 (pow x -0.16666666666666666))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(sqrt(x));
	double tmp;
	if (x <= 1.65e+248) {
		tmp = 0.3333333333333333 * (t_0 * cbrt((t_0 * (1.0 / x))));
	} else {
		tmp = (1.0 / sqrt(x)) * (0.3333333333333333 * pow(x, -0.16666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.sqrt(Math.sqrt(x));
	double tmp;
	if (x <= 1.65e+248) {
		tmp = 0.3333333333333333 * (t_0 * Math.cbrt((t_0 * (1.0 / x))));
	} else {
		tmp = (1.0 / Math.sqrt(x)) * (0.3333333333333333 * Math.pow(x, -0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(sqrt(x)))
	tmp = 0.0
	if (x <= 1.65e+248)
		tmp = Float64(0.3333333333333333 * Float64(t_0 * cbrt(Float64(t_0 * Float64(1.0 / x)))));
	else
		tmp = Float64(Float64(1.0 / sqrt(x)) * Float64(0.3333333333333333 * (x ^ -0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[Sqrt[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.65e+248], N[(0.3333333333333333 * N[(t$95$0 * N[Power[N[(t$95$0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Power[x, -0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6500000000000001e248

   1. Initial program 7.6%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 58.4

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

   5. Simplified 58.4%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 95.9

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

   7. Applied egg-rr 95.9%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 96.0%

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

   11. Applied egg-rr 96.1%

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

   if 1.6500000000000001e248 < 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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 5.2

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

   5. Simplified 5.2%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.3

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

   7. Applied egg-rr 98.3%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.2%

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

   11. Applied egg-rr 91.9%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(0.3333333333333333 \cdot {x}^{-0.16666666666666666}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+248}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{1}{\sqrt{\sqrt{x}}} \cdot \sqrt[3]{\frac{1}{\sqrt{\sqrt{x}}} \cdot \frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left(0.3333333333333333 \cdot {x}^{-0.16666666666666666}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{1}{\sqrt{x}}}{{x}^{0.16666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+154)
   (* 0.3333333333333333 (cbrt (* (/ 1.0 x) (/ 1.0 x))))
   (* 0.3333333333333333 (/ (/ 1.0 (sqrt x)) (pow x 0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * ((1.0 / sqrt(x)) / pow(x, 0.16666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * ((1.0 / Math.sqrt(x)) / Math.pow(x, 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+154)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) * Float64(1.0 / x))));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64(1.0 / sqrt(x)) / (x ^ 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+154], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / N[Power[x, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e154

   1. Initial program 9.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 4.00000000000000015e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.5%

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

       1. cbrt-undiv N/A

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

       2. pow1/3 N/A

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

       3. div-inv N/A

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

       4. neg-mul-1 N/A

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

       5. times-frac N/A

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

       6. unpow-prod-down N/A

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

       7. associate-/l/ N/A

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

       8. associate-/r* N/A

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

       9. metadata-eval N/A

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

       10. inv-pow N/A

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

       11. rem-square-sqrt N/A

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

       12. pow2 N/A

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

       13. pow-div N/A

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

       14. metadata-eval N/A

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

       15. metadata-eval N/A

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

       16. metadata-eval N/A

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

       17. sqrt-pow2 N/A

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

       18. pow-pow N/A

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

       19. inv-pow N/A

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

   11. Applied egg-rr 92.4%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{\sqrt{x}}}{{x}^{0.16666666666666666}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x}} \cdot \left(0.3333333333333333 \cdot {x}^{-0.16666666666666666}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+154)
   (* 0.3333333333333333 (cbrt (* (/ 1.0 x) (/ 1.0 x))))
   (* (/ 1.0 (sqrt x)) (* 0.3333333333333333 (pow x -0.16666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = (1.0 / sqrt(x)) * (0.3333333333333333 * pow(x, -0.16666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = (1.0 / Math.sqrt(x)) * (0.3333333333333333 * Math.pow(x, -0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+154)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) * Float64(1.0 / x))));
	else
		tmp = Float64(Float64(1.0 / sqrt(x)) * Float64(0.3333333333333333 * (x ^ -0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+154], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Power[x, -0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e154

   1. Initial program 9.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 4.00000000000000015e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.5%

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

   11. Applied egg-rr 92.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x}} \cdot \left(0.3333333333333333 \cdot {x}^{-0.16666666666666666}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-0.16666666666666666}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+154)
   (* 0.3333333333333333 (cbrt (* (/ 1.0 x) (/ 1.0 x))))
   (* 0.3333333333333333 (/ (pow x -0.16666666666666666) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * (pow(x, -0.16666666666666666) / sqrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -0.16666666666666666) / Math.sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+154)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) * Float64(1.0 / x))));
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -0.16666666666666666) / sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+154], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e154

   1. Initial program 9.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. /-lowering-/.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 4.00000000000000015e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   9. Applied egg-rr 92.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-0.16666666666666666}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 93.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{{x}^{-0.16666666666666666}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+154)
   (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x)))
   (* 0.3333333333333333 (/ (pow x -0.16666666666666666) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 * (pow(x, -0.16666666666666666) / sqrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4e+154) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 * (Math.pow(x, -0.16666666666666666) / Math.sqrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+154)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) / x)));
	else
		tmp = Float64(0.3333333333333333 * Float64((x ^ -0.16666666666666666) / sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+154], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e154

   1. Initial program 9.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 4.00000000000000015e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

   9. Applied egg-rr 92.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{{x}^{-0.16666666666666666}}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+155}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\frac{1}{x}\right)}^{0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+155)
   (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x)))
   (* 0.3333333333333333 (pow (/ 1.0 x) 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+155) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 * pow((1.0 / x), 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4e+155) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 * Math.pow((1.0 / x), 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+155)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) / x)));
	else
		tmp = Float64(0.3333333333333333 * (Float64(1.0 / x) ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+155], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[(1.0 / x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.40000000000000008e155

   1. Initial program 9.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 94.5

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

   7. Applied egg-rr 94.5%

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

   if 1.40000000000000008e155 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.5%

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

       1. cbrt-undiv N/A

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

       2. pow1/3 N/A

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

       3. associate-/l/ N/A

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

       4. rem-square-sqrt N/A

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

       5. div-inv N/A

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

       6. associate-*l/ N/A

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

       7. metadata-eval N/A

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

       8. frac-2neg N/A

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

       9. pow2 N/A

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

       10. pow-pow N/A

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

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

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

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

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

       13. metadata-eval 89.3

           \[\leadsto 0.3333333333333333 \cdot {\left(\frac{1}{x}\right)}^{\color{blue}{0.6666666666666666}} \]

   11. Applied egg-rr 89.3%

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

Alternative 13: 92.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* 0.3333333333333333 (cbrt (/ 1.0 (* x x))))
   (* 0.3333333333333333 (pow (/ 1.0 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((1.0 / 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((1.0 / 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 * (Float64(1.0 / 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[N[(1.0 / x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.8%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 94.3

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

   5. Simplified 94.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 4.7

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

   5. Simplified 4.7%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. cbrt-div N/A

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

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

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

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

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. neg-lowering-neg.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. rem-square-sqrt N/A

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

      2. associate-/r* N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. pow2 N/A

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

      10. sqr-neg N/A

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

      11. rem-square-sqrt N/A

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

      12. pow-pow N/A

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

      13. pow1/2 N/A

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

      14. inv-pow N/A

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

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

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

   9. Applied egg-rr 98.6%

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

       1. cbrt-undiv N/A

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

       2. pow1/3 N/A

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

       3. associate-/l/ N/A

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

       4. rem-square-sqrt N/A

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

       5. div-inv N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\frac{\color{blue}{-1 \cdot \frac{1}{x}}}{\mathsf{neg}\left(x\right)}\right)}^{\frac{1}{3}} \]

       6. associate-*l/ N/A

          \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\frac{-1}{\mathsf{neg}\left(x\right)} \cdot \frac{1}{x}\right)}}^{\frac{1}{3}} \]

       7. metadata-eval N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{\mathsf{neg}\left(x\right)} \cdot \frac{1}{x}\right)}^{\frac{1}{3}} \]

       8. frac-2neg N/A

          \[\leadsto \frac{1}{3} \cdot {\left(\color{blue}{\frac{1}{x}} \cdot \frac{1}{x}\right)}^{\frac{1}{3}} \]

       9. pow2 N/A

          \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left({\left(\frac{1}{x}\right)}^{2}\right)}}^{\frac{1}{3}} \]

       10. pow-pow N/A

           \[\leadsto \frac{1}{3} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{\left(2 \cdot \frac{1}{3}\right)}} \]

       11. pow-lowering-pow.f64 N/A

           \[\leadsto \frac{1}{3} \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{\left(2 \cdot \frac{1}{3}\right)}} \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \frac{1}{3} \cdot {\color{blue}{\left(\frac{1}{x}\right)}}^{\left(2 \cdot \frac{1}{3}\right)} \]

       13. metadata-eval 89.3

           \[\leadsto 0.3333333333333333 \cdot {\left(\frac{1}{x}\right)}^{\color{blue}{0.6666666666666666}} \]

   11. Applied egg-rr 89.3%

       \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 / pow(x, 0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 / (x ** 0.6666666666666666d0)
end function

Java

public static double code(double x) {
	return 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 / math.pow(x, 0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 / (x ^ 0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{-1 \cdot -1}{{x}^{2}}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{\color{blue}{1}}{{x}^{2}}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

   9. *-lowering-*.f64 48.5

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

5. Simplified 48.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Step-by-step derivation

   1. cbrt-div N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{x \cdot x}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \frac{\color{blue}{1}}{\sqrt[3]{x \cdot x}} \]

   3. cbrt-prod N/A

      \[\leadsto \frac{1}{3} \cdot \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \]

   6. pow1/3 N/A

      \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{\frac{1}{3}}} \cdot \sqrt[3]{x}} \]

   7. pow1/3 N/A

      \[\leadsto \frac{\frac{1}{3}}{{x}^{\frac{1}{3}} \cdot \color{blue}{{x}^{\frac{1}{3}}}} \]

   8. pow-sqr N/A

      \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{\left(2 \cdot \frac{1}{3}\right)}}} \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{3}}{\color{blue}{{x}^{\left(2 \cdot \frac{1}{3}\right)}}} \]

   10. metadata-eval 88.6

       \[\leadsto \frac{0.3333333333333333}{{x}^{\color{blue}{0.6666666666666666}}} \]

7. Applied egg-rr 88.6%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 15: 88.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (pow x -0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 * pow(x, -0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (x ** (-0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 * math.pow(x, -0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 * (x ^ -0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (x ^ -0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{\color{blue}{-1 \cdot -1}}{{x}^{2}}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{-1 \cdot \frac{-1}{{x}^{2}}}} \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{-1 \cdot -1}{{x}^{2}}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{\color{blue}{1}}{{x}^{2}}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\color{blue}{\frac{1}{{x}^{2}}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{3} \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

   9. *-lowering-*.f64 48.5

      \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{\color{blue}{x \cdot x}}} \]

5. Simplified 48.5%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{1}{3}} \]

   3. pow1/3 N/A

      \[\leadsto \color{blue}{{\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}}} \cdot \frac{1}{3} \]

   4. inv-pow N/A

      \[\leadsto {\color{blue}{\left({\left(x \cdot x\right)}^{-1}\right)}}^{\frac{1}{3}} \cdot \frac{1}{3} \]

   5. pow-pow N/A

      \[\leadsto \color{blue}{{\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)}} \cdot \frac{1}{3} \]

   6. pow2 N/A

      \[\leadsto {\color{blue}{\left({x}^{2}\right)}}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{1}{3} \]

   7. pow-pow N/A

      \[\leadsto \color{blue}{{x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}} \cdot \frac{1}{3} \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto \color{blue}{{x}^{\left(2 \cdot \left(-1 \cdot \frac{1}{3}\right)\right)}} \cdot \frac{1}{3} \]

   9. metadata-eval N/A

      \[\leadsto {x}^{\left(2 \cdot \color{blue}{\frac{-1}{3}}\right)} \cdot \frac{1}{3} \]

   10. metadata-eval 88.6

       \[\leadsto {x}^{\color{blue}{-0.6666666666666666}} \cdot 0.3333333333333333 \]

7. Applied egg-rr 88.6%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]

8. Final simplification 88.6%

   \[\leadsto 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \]
9. Add Preprocessing

Alternative 16: 5.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\sqrt[3]{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cbrt (- x))))

C

double code(double x) {
	return -cbrt(-x);
}

Java

public static double code(double x) {
	return -Math.cbrt(-x);
}

Julia

function code(x)
	return Float64(-cbrt(Float64(-x)))
end

Wolfram

code[x_] := (-N[Power[(-x), 1/3], $MachinePrecision])

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]

   2. cbrt-lowering-cbrt.f64 1.8

      \[\leadsto 1 - \color{blue}{\sqrt[3]{x}} \]

5. Simplified 1.8%

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \sqrt[3]{x}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   3. cbrt-lowering-cbrt.f64 1.8

      \[\leadsto -\color{blue}{\sqrt[3]{x}} \]

8. Simplified 1.8%

   \[\leadsto \color{blue}{-\sqrt[3]{x}} \]
9. Step-by-step derivation

   1. pow1/3 N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{{x}^{\frac{1}{3}}}\right) \]

   2. pow-lowering-pow.f64 1.8

      \[\leadsto -\color{blue}{{x}^{0.3333333333333333}} \]

10. Applied egg-rr 1.8%

    \[\leadsto -\color{blue}{{x}^{0.3333333333333333}} \]
11. Step-by-step derivation

    1. sqr-pow N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{x}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right) \]

    2. pow-prod-down N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(x \cdot x\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right) \]

    3. sqr-neg N/A

       \[\leadsto \mathsf{neg}\left({\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \]

    4. pow-prod-down N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}}\right) \]

    5. sqr-pow N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\frac{1}{3}}}\right) \]

    6. pow1/3 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}\right) \]

    7. cbrt-lowering-cbrt.f64 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}\right) \]

    8. neg-lowering-neg.f64 5.3

       \[\leadsto -\sqrt[3]{\color{blue}{-x}} \]

12. Applied egg-rr 5.3%

    \[\leadsto -\color{blue}{\sqrt[3]{-x}} \]
13. Add Preprocessing

Alternative 17: 1.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (cbrt x)))

C

double code(double x) {
	return 1.0 - cbrt(x);
}

Java

public static double code(double x) {
	return 1.0 - Math.cbrt(x);
}

Julia

function code(x)
	return Float64(1.0 - cbrt(x))
end

Wolfram

code[x_] := N[(1.0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]

   2. cbrt-lowering-cbrt.f64 1.8

      \[\leadsto 1 - \color{blue}{\sqrt[3]{x}} \]

5. Simplified 1.8%

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
6. Add Preprocessing

Alternative 18: 1.8% 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 Float64(-cbrt(x))
end

Wolfram

code[x_] := (-N[Power[x, 1/3], $MachinePrecision])

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]

   2. cbrt-lowering-cbrt.f64 1.8

      \[\leadsto 1 - \color{blue}{\sqrt[3]{x}} \]

5. Simplified 1.8%

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \sqrt[3]{x}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt[3]{x}\right)} \]

   3. cbrt-lowering-cbrt.f64 1.8

      \[\leadsto -\color{blue}{\sqrt[3]{x}} \]

8. Simplified 1.8%

   \[\leadsto \color{blue}{-\sqrt[3]{x}} \]
9. 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 2024199 
(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)))