2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.5%

Time: 8.5s

Alternatives: 14

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: 6.9% 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: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({\left(\sqrt[3]{x}\right)}^{2} + {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 6e-11)
   (/ (pow (cbrt x) -2.0) 3.0)
   (/
    (- (+ 1.0 x) x)
    (+
     (cbrt (fma x x x))
     (+ (pow (cbrt x) 2.0) (pow (exp 0.6666666666666666) (log1p x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 6e-11) {
		tmp = pow(cbrt(x), -2.0) / 3.0;
	} else {
		tmp = ((1.0 + x) - x) / (cbrt(fma(x, x, x)) + (pow(cbrt(x), 2.0) + pow(exp(0.6666666666666666), log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 6e-11)
		tmp = Float64((cbrt(x) ^ -2.0) / 3.0);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(cbrt(fma(x, x, x)) + Float64((cbrt(x) ^ 2.0) + (exp(0.6666666666666666) ^ log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 6e-11], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Exp[0.6666666666666666], $MachinePrecision], N[Log[1 + x], $MachinePrecision]], $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)) < 6e-11

   1. Initial program 4.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.5%

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

   if 6e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 65.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 63.6

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

   4. Applied rewrites 63.6%

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

   6. Applied rewrites 97.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\left({\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} + {\left(\sqrt[3]{x}\right)}^{2}\right) - \left(-\sqrt[3]{\mathsf{fma}\left(x, x, x\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({\left(\sqrt[3]{x}\right)}^{2} + {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_0, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 6e-11)
     (/ (pow (cbrt x) -2.0) 3.0)
     (/
      (- (+ 1.0 x) x)
      (fma
       (cbrt x)
       (+ (cbrt x) t_0)
       (pow (exp 0.6666666666666666) (log1p x)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 6e-11) {
		tmp = pow(cbrt(x), -2.0) / 3.0;
	} else {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt(x) + t_0), pow(exp(0.6666666666666666), log1p(x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 6e-11)
		tmp = Float64((cbrt(x) ^ -2.0) / 3.0);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(x) + t_0), (exp(0.6666666666666666) ^ log1p(x))));
	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], 6e-11], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[N[Exp[0.6666666666666666], $MachinePrecision], N[Log[1 + x], $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)) < 6e-11

   1. Initial program 4.3%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. *-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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 51.5

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

   5. Applied rewrites 51.5%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 98.5%

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

   if 6e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 65.7%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 63.6

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

   4. Applied rewrites 63.6%

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

   6. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 6 \cdot 10^{-11}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 30000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.25}, -{x}^{0.08333333333333333}, \sqrt[3]{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 30000000.0)
   (fma (pow x 0.25) (- (pow x 0.08333333333333333)) (cbrt (+ 1.0 x)))
   (/ (pow (cbrt x) -2.0) 3.0)))

C

double code(double x) {
	double tmp;
	if (x <= 30000000.0) {
		tmp = fma(pow(x, 0.25), -pow(x, 0.08333333333333333), cbrt((1.0 + x)));
	} else {
		tmp = pow(cbrt(x), -2.0) / 3.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 30000000.0)
		tmp = fma((x ^ 0.25), Float64(-(x ^ 0.08333333333333333)), cbrt(Float64(1.0 + x)));
	else
		tmp = Float64((cbrt(x) ^ -2.0) / 3.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 30000000.0], N[(N[Power[x, 0.25], $MachinePrecision] * (-N[Power[x, 0.08333333333333333], $MachinePrecision]) + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e7

   1. Initial program 81.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-pow N/A

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

      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. lift-cbrt.f64 N/A

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

      11. pow1/3 N/A

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

      12. metadata-eval N/A

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

      13. pow-sqr N/A

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

      14. lift-pow.f64 N/A

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

      15. sqr-pow N/A

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

      16. associate-*r* N/A

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

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

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

      18. lift-cbrt.f64 N/A

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

      19. pow1/3 N/A

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

      20. pow-to-exp N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + \color{blue}{e^{\log \left(x + 1\right) \cdot \frac{1}{3}}} \]

      21. lift-+.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + e^{\log \color{blue}{\left(x + 1\right)} \cdot \frac{1}{3}} \]

      22. +-commutative N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + e^{\log \color{blue}{\left(1 + x\right)} \cdot \frac{1}{3}} \]

      23. lift-log1p.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + e^{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot \frac{1}{3}} \]

      24. *-commutative N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + e^{\color{blue}{\frac{1}{3} \cdot \mathsf{log1p}\left(x\right)}} \]

      25. pow-exp N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + \color{blue}{{\left(e^{\frac{1}{3}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} \]

      26. lift-exp.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + {\color{blue}{\left(e^{\frac{1}{3}}\right)}}^{\left(\mathsf{log1p}\left(x\right)\right)} \]

      27. lift-pow.f64 N/A

          \[\leadsto \left({x}^{\frac{1}{6}} \cdot {x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right) \cdot \left(\mathsf{neg}\left({x}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right) + \color{blue}{{\left(e^{\frac{1}{3}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} \]

   6. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, -{x}^{0.08333333333333333}, \sqrt[3]{x + 1}\right)} \]

   if 3e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30000000:\\ \;\;\;\;\mathsf{fma}\left({x}^{0.25}, -{x}^{0.08333333333333333}, \sqrt[3]{1 + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 46000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (pow (pow x 0.16666666666666666) 2.0))
   (/ (pow (cbrt x) -2.0) 3.0)))

C

double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - pow(pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = pow(cbrt(x), -2.0) / 3.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 46000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.pow(Math.pow(x, 0.16666666666666666), 2.0);
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) / 3.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 46000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - ((x ^ 0.16666666666666666) ^ 2.0));
	else
		tmp = Float64((cbrt(x) ^ -2.0) / 3.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 46000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[N[Power[x, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e7

   1. Initial program 81.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 83.1

         \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   6. Applied rewrites 83.1%

      \[\leadsto \color{blue}{{\left(x + 1\right)}^{0.3333333333333333}} - {\left({x}^{0.16666666666666666}\right)}^{2} \]

   if 4.6e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 46000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - {\left({x}^{0.16666666666666666}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 24000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{x}\right)}^{-2}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 24000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (cbrt x))
   (/ (pow (cbrt x) -2.0) 3.0)))

C

double code(double x) {
	double tmp;
	if (x <= 24000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = pow(cbrt(x), -2.0) / 3.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 24000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) / 3.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 24000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64((cbrt(x) ^ -2.0) / 3.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 24000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] / 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e7

   1. Initial program 81.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-log1p.f64 80.6

          \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot 0.3333333333333333} - \sqrt[3]{x} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{3}}} - \sqrt[3]{x} \]

      2. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{3}}} - \sqrt[3]{x} \]

      3. lift-log1p.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if 2.4e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 98.2%

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

Alternative 6: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 24000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 24000000.0)
   (- (pow (+ 1.0 x) 0.3333333333333333) (cbrt x))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 24000000.0) {
		tmp = pow((1.0 + x), 0.3333333333333333) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 24000000.0) {
		tmp = Math.pow((1.0 + x), 0.3333333333333333) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 24000000.0)
		tmp = Float64((Float64(1.0 + x) ^ 0.3333333333333333) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 24000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.3333333333333333], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e7

   1. Initial program 81.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-log1p.f64 80.6

          \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right)} \cdot 0.3333333333333333} - \sqrt[3]{x} \]

   4. Applied rewrites 80.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot \frac{1}{3}}} - \sqrt[3]{x} \]

      2. lift-*.f64 N/A

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{1}{3}}} - \sqrt[3]{x} \]

      3. lift-log1p.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if 2.4e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 24000000:\\ \;\;\;\;{\left(1 + x\right)}^{0.3333333333333333} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 16000000:\\ \;\;\;\;\sqrt[3]{1 + x} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 16000000.0)
   (- (cbrt (+ 1.0 x)) (pow x 0.3333333333333333))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 16000000.0) {
		tmp = cbrt((1.0 + x)) - pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 16000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.pow(x, 0.3333333333333333);
	} else {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 16000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - (x ^ 0.3333333333333333));
	else
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 16000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6e7

   1. Initial program 81.6%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. lower-pow.f64 81.8

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

   4. Applied rewrites 81.8%

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

   if 1.6e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 16000000:\\ \;\;\;\;\sqrt[3]{1 + x} - {x}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 32500000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 32500000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (* 0.3333333333333333 (pow (cbrt x) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * pow(cbrt(x), -2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 32500000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = 0.3333333333333333 * Math.pow(Math.cbrt(x), -2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 32500000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * (cbrt(x) ^ -2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 32500000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.25e7

   1. Initial program 81.6%

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

   if 3.25e7 < x

   1. Initial program 5.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 52.4

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

   5. Applied rewrites 52.4%

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

   7. Applied rewrites 98.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 32500000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% 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 8.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 51.6

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

5. Applied rewrites 51.6%

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

7. Applied rewrites 95.2%

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

8. Final simplification 95.2%

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

Alternative 10: 92.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/ 1.0 (* (cbrt (* x x)) 3.0))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (cbrt((x * x)) * 3.0);
	} 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 = 1.0 / (Math.cbrt((x * x)) * 3.0);
	} 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(1.0 / Float64(cbrt(Float64(x * x)) * 3.0));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 12.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 92.6%

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

   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 9.0

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

   5. Applied rewrites 9.0%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 89.1%

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

Alternative 11: 92.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 12.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 92.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. *-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. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 9.0

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

   5. Applied rewrites 9.0%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 89.1%

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

Alternative 12: 88.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (pow x 0.6666666666666666)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(0.3333333333333333 / (x ^ 0.6666666666666666))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 51.6

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

5. Applied rewrites 51.6%

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

7. Applied rewrites 95.2%

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

9. Applied rewrites 87.7%

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

Alternative 13: 88.8% 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 8.6%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. *-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. unpow2 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 51.6

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

5. Applied rewrites 51.6%

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

7. Applied rewrites 87.7%

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

Alternative 14: 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 8.6%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow-to-exp N/A

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

   5. pow-exp N/A

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

   6. *-commutative N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. pow-to-exp N/A

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

   12. lift-cbrt.f64 N/A

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

   13. rem-cube-cbrt N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-log1p.f64 7.4

       \[\leadsto {\left(e^{0.3333333333333333}\right)}^{\color{blue}{\left(\mathsf{log1p}\left(x\right)\right)}} - \sqrt[3]{x} \]

4. Applied rewrites 7.4%

   \[\leadsto \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} - \sqrt[3]{x} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{0} \]
6. Step-by-step derivation

7. Applied rewrites 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 2024295 
(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)))