2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.4%

Time: 9.8s

Alternatives: 13

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: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{{x}^{0.6666666666666666} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6e+14)
   (/
    (+ x (- 1.0 x))
    (+
     (pow x 0.6666666666666666)
     (+ (cbrt (fma x x x)) (cbrt (* (+ 1.0 x) (+ 1.0 x))))))
   (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 6e+14) {
		tmp = (x + (1.0 - x)) / (pow(x, 0.6666666666666666) + (cbrt(fma(x, x, x)) + cbrt(((1.0 + x) * (1.0 + x)))));
	} else {
		tmp = (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6e+14)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64((x ^ 0.6666666666666666) + Float64(cbrt(fma(x, x, x)) + cbrt(Float64(Float64(1.0 + x) * Float64(1.0 + x))))));
	else
		tmp = Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6e+14], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e14

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 67.4

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

   4. Applied egg-rr 67.4%

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

   6. Applied egg-rr 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 97.7

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

      3. rem-cbrt-cube N/A

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

      4. lower-cbrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 98.8

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

   8. Applied egg-rr 98.8%

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

   if 6e14 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.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. lower-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. lower-/.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. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. lift-cbrt.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{{x}^{0.6666666666666666} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \sqrt[3]{\left(1 + x\right) \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{x \cdot x} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 5e-11)
   (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x))
   (/
    (+ x (- 1.0 x))
    (+
     (cbrt (* x x))
     (+ (cbrt (fma x x x)) (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 5e-11) {
		tmp = (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
	} else {
		tmp = (x + (1.0 - x)) / (cbrt((x * x)) + (cbrt(fma(x, x, x)) + pow((1.0 + x), 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 5e-11)
		tmp = Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x));
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(cbrt(Float64(x * x)) + Float64(cbrt(fma(x, x, x)) + (Float64(1.0 + x) ^ 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.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. lower-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. lower-/.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. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. lift-cbrt.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 67.4

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

   4. Applied egg-rr 67.4%

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

   6. Applied egg-rr 97.7%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. lower-*.f64 98.5

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{x \cdot x} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({x}^{0.6666666666666666} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 5e-11)
   (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x))
   (/
    (+ x (- 1.0 x))
    (+
     (cbrt (fma x x x))
     (+ (pow x 0.6666666666666666) (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 5e-11) {
		tmp = (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
	} else {
		tmp = (x + (1.0 - x)) / (cbrt(fma(x, x, x)) + (pow(x, 0.6666666666666666) + pow((1.0 + x), 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 5e-11)
		tmp = Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x));
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(cbrt(fma(x, x, x)) + Float64((x ^ 0.6666666666666666) + (Float64(1.0 + x) ^ 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.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. lower-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. lower-/.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. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. lift-cbrt.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 67.4

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

   4. Applied egg-rr 67.4%

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

   6. Applied egg-rr 97.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. associate-+r+ N/A

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

      9. lower-+.f64 N/A

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

      10. lower-+.f64 97.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 97.9

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({x}^{0.6666666666666666} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 5e-11)
   (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x))
   (/
    1.0
    (+
     (pow x 0.6666666666666666)
     (+ (cbrt (fma x x x)) (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 5e-11) {
		tmp = (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
	} else {
		tmp = 1.0 / (pow(x, 0.6666666666666666) + (cbrt(fma(x, x, x)) + pow((1.0 + x), 0.6666666666666666)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 5e-11)
		tmp = Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x));
	else
		tmp = Float64(1.0 / Float64((x ^ 0.6666666666666666) + Float64(cbrt(fma(x, x, x)) + (Float64(1.0 + x) ^ 0.6666666666666666))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 0.6666666666666666], $MachinePrecision] + N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.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. lower-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. lower-/.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. lower-*.f64 49.9

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

   5. Simplified 49.9%

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

      1. associate-/r* N/A

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

      2. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 98.5

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

   7. Applied egg-rr 98.5%

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

      1. lift-cbrt.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 98.6

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

   9. Applied egg-rr 98.6%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 67.4%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 67.4

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

   4. Applied egg-rr 67.4%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in x around 0

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

   9. Simplified 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. lift-+.f64 N/A

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

   2. lift-cbrt.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. flip-- N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. flip-- N/A

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

   9. lift--.f64 N/A

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

   10. lower-/.f64 7.6

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

4. Applied egg-rr 7.6%

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

6. Applied egg-rr 9.7%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

9. Simplified 97.8%

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

Alternative 6: 96.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x)))

C

double code(double x) {
	return (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
}

Java

public static double code(double x) {
	return (1.0 / (Math.cbrt(x) * 3.0)) / Math.cbrt(x);
}

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x))
end

Wolfram

code[x_] := N[(N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

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. lower-*.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. lower-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. lower-/.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. lower-*.f64 50.0

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

5. Simplified 50.0%

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

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. associate-/r* N/A

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

   8. un-div-inv N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 96.1

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

7. Applied egg-rr 96.1%

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

   1. lift-cbrt.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 96.2

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

9. Applied egg-rr 96.2%

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

Alternative 7: 96.3% 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 7.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.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. lower-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. lower-/.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. lower-*.f64 50.0

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

5. Simplified 50.0%

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

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

   3. cbrt-div N/A

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

   4. metadata-eval N/A

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

   5. lift-cbrt.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. associate-/r* N/A

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

   8. inv-pow N/A

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

   9. pow2 N/A

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

   10. pow-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval 96.1

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

7. Applied egg-rr 96.1%

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

Alternative 8: 92.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{0.6666666666666666}}{0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (/ 1.0 (/ (pow x 0.6666666666666666) 0.3333333333333333))))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = 1.0 / (pow(x, 0.6666666666666666) / 0.3333333333333333);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 1.0 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = 1.0 / (Math.pow(x, 0.6666666666666666) / 0.3333333333333333);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(1.0 / Float64((x ^ 0.6666666666666666) / 0.3333333333333333));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 0.6666666666666666], $MachinePrecision] / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 10.3%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 10.3

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

   4. Applied egg-rr 10.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 94.4

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

   7. Simplified 94.4%

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

   if 1.31999999999999998e154 < 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. lower-*.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. lower-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. lower-/.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. lower-*.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. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. un-div-inv N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. pow1/3 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. pow1/3 N/A

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

      16. pow-prod-up N/A

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

      17. lower-pow.f64 N/A

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

      18. metadata-eval 89.2

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

   7. Applied egg-rr 89.2%

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

Alternative 9: 92.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{3 \cdot \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (/ 1.0 (* 3.0 (cbrt (* x x))))
   (* 0.3333333333333333 (/ 1.0 (pow x 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 1.0 / (3.0 * cbrt((x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / pow(x, 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 1.0 / (3.0 * Math.cbrt((x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / Math.pow(x, 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+154)
		tmp = Float64(1.0 / Float64(3.0 * cbrt(Float64(x * x))));
	else
		tmp = Float64(0.3333333333333333 * Float64(1.0 / (x ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.32e+154], N[(1.0 / N[(3.0 * N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 10.3%

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

      1. lift-+.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. flip-- N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. flip-- N/A

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

      9. lift--.f64 N/A

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

      10. lower-/.f64 10.3

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

   4. Applied egg-rr 10.3%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cbrt.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 94.4

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

   7. Simplified 94.4%

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

   if 1.31999999999999998e154 < 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. lower-*.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. lower-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. lower-/.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. lower-*.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. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. pow1/3 N/A

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

      11. lift-cbrt.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 89.2

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

   7. Applied egg-rr 89.2%

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

Alternative 10: 91.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.32e+154)
   (* 0.3333333333333333 (cbrt (/ 1.0 (* x x))))
   (* 0.3333333333333333 (/ 1.0 (pow x 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / (x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / pow(x, 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.32e+154) {
		tmp = 0.3333333333333333 * Math.cbrt((1.0 / (x * x)));
	} else {
		tmp = 0.3333333333333333 * (1.0 / Math.pow(x, 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.32e+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, 1.32e+154], N[(0.3333333333333333 * N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 10.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lower-*.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. lower-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. lower-/.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. lower-*.f64 94.0

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

   5. Simplified 94.0%

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

   if 1.31999999999999998e154 < 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. lower-*.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. lower-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. lower-/.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. lower-*.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. cbrt-div N/A

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

      3. cbrt-div N/A

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

      4. metadata-eval N/A

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

      5. lift-cbrt.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. pow1/3 N/A

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

      11. lift-cbrt.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow-prod-up N/A

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

      14. lower-pow.f64 N/A

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

      15. metadata-eval 89.2

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

   7. Applied egg-rr 89.2%

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

Alternative 11: 88.7% 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 7.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. lower-*.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. lower-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. lower-/.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. lower-*.f64 50.0

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

5. Simplified 50.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 50.0

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-pow N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. pow-pow N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval 88.4

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

7. Applied egg-rr 88.4%

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

8. Final simplification 88.4%

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

Alternative 12: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. lift-cbrt.f64 7.6

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

   2. rem-exp-log N/A

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

   3. lift-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. log-pow N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-log.f64 7.7

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

4. Applied egg-rr 7.7%

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

5. Taylor expanded in x around inf

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

   1. lower-cbrt.f64 5.4

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

7. Simplified 5.4%

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

Alternative 13: 4.1% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.6%

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

   1. lift-cbrt.f64 7.6

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

   2. rem-exp-log N/A

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

   3. lift-cbrt.f64 N/A

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

   4. pow1/3 N/A

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

   5. log-pow N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-log.f64 7.7

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

4. Applied egg-rr 7.7%

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

5. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

      \[\leadsto x \cdot \color{blue}{0} \]

   4. mul0-rgt 4.1

      \[\leadsto \color{blue}{0} \]

7. Simplified 4.1%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024207 
(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)))