2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.9%

Time: 9.5s

Alternatives: 11

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: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 93.4%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{0.16666666666666666}}}{\color{blue}{\sqrt{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{0.5}}}{\sqrt{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.0%

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

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

   1. Initial program 57.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. pow1/3 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval 54.8

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

   4. Applied rewrites 54.8%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/3 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. flip3-+ N/A

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

      11. lower-/.f64 N/A

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

   6. Applied rewrites 97.0%

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

4. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   7. Applied rewrites 93.4%

      \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{0.16666666666666666}}}{\color{blue}{\sqrt{x}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.0%

      \[\leadsto \frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{0.5}}}{\sqrt{x}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.0%

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

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

   1. Initial program 57.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. pow1/3 N/A

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

      9. pow-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. unpow1/2 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval 54.8

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

   4. Applied rewrites 54.8%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. metadata-eval N/A

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

      6. pow1/3 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. flip3-- N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. rem-cube-cbrt N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

   6. Applied rewrites 96.9%

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

4. Final simplification 98.9%

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

Alternative 3: 97.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 0.3333333333333333 (sqrt (cbrt x))) (sqrt x)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 91.9%

   \[\leadsto \frac{\frac{0.3333333333333333}{{x}^{0.16666666666666666}}}{\color{blue}{\sqrt{x}}} \]
8. Step-by-step derivation

9. Applied rewrites 97.3%

   \[\leadsto \frac{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{0.5}}}{\sqrt{x}} \]
10. Step-by-step derivation

11. Applied rewrites 97.3%

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

Alternative 4: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (* (/ 1.0 (sqrt x)) (pow x -0.16666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = ((1.0 / sqrt(x)) * pow(x, -0.16666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = ((1.0 / Math.sqrt(x)) * Math.pow(x, -0.16666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(1.0 / sqrt(x)) * (x ^ -0.16666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.16666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 92.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\sqrt{x}} \cdot {x}^{-0.16666666666666666}\right) \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{1}{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.16666666666666666}}{\sqrt{x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (cbrt (/ 1.0 (* x x))) 0.3333333333333333)
   (* (/ (pow x -0.16666666666666666) (sqrt x)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (pow(x, -0.16666666666666666) / sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (Math.pow(x, -0.16666666666666666) / Math.sqrt(x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64((x ^ -0.16666666666666666) / sqrt(x)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(N[Power[x, -0.16666666666666666], $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 98.5%

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

   9. Applied rewrites 92.2%

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

Alternative 6: 91.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

   7. Applied rewrites 89.1%

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

Alternative 7: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (sqrt x) -1.3333333333333333) 0.3333333333333333))

C

double code(double x) {
	return pow(sqrt(x), -1.3333333333333333) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (sqrt(x) ** (-1.3333333333333333d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(Math.sqrt(x), -1.3333333333333333) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(math.sqrt(x), -1.3333333333333333) * 0.3333333333333333

Julia

function code(x)
	return Float64((sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (sqrt(x) ^ -1.3333333333333333) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[N[Sqrt[x], $MachinePrecision], -1.3333333333333333], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 89.0%

   \[\leadsto {\left(\sqrt{x}\right)}^{-1.3333333333333333} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 8: 88.7% 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 6.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 89.0%

   \[\leadsto \frac{0.3333333333333333}{\color{blue}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 9: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.6666666666666666} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow x -0.6666666666666666) 0.3333333333333333))

C

double code(double x) {
	return pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.6666666666666666d0)) * 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.6666666666666666) * 0.3333333333333333;
}

Python

def code(x):
	return math.pow(x, -0.6666666666666666) * 0.3333333333333333

Julia

function code(x)
	return Float64((x ^ -0.6666666666666666) * 0.3333333333333333)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.6666666666666666) * 0.3333333333333333;
end

Wolfram

code[x_] := N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. associate-*r/ N/A

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

   5. lower-cbrt.f64 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 52.2

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

5. Applied rewrites 52.2%

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

7. Applied rewrites 89.0%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
8. Add Preprocessing

Alternative 10: 5.4% 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(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 6.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 1.8

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

7. Applied rewrites 1.8%

   \[\leadsto 1 - \color{blue}{{x}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow2 N/A

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

   5. sqr-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. pow-prod-down N/A

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

   9. pow-prod-up N/A

      \[\leadsto 1 - \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}} \]

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

   12. lift-cbrt.f64 5.3

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

9. Applied rewrites 5.3%

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

Alternative 11: 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 6.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
6. 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 2024235 
(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)))