2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.6%

Time: 11.0s

Alternatives: 16

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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, x - -1\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x - -1} \cdot \sqrt[3]{x} + {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 0.0)
   (* 0.3333333333333333 (* (/ (pow x -0.25) (cbrt x)) (pow (cbrt x) -0.25)))
   (/
    (fma -1.0 x (- x -1.0))
    (fma
     (cbrt x)
     (cbrt x)
     (+
      (* (cbrt (- x -1.0)) (cbrt x))
      (pow (exp 0.6666666666666666) (log1p x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * ((pow(x, -0.25) / cbrt(x)) * pow(cbrt(x), -0.25));
	} else {
		tmp = fma(-1.0, x, (x - -1.0)) / fma(cbrt(x), cbrt(x), ((cbrt((x - -1.0)) * cbrt(x)) + pow(exp(0.6666666666666666), log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 0.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64((x ^ -0.25) / cbrt(x)) * (cbrt(x) ^ -0.25)));
	else
		tmp = Float64(fma(-1.0, x, Float64(x - -1.0)) / fma(cbrt(x), cbrt(x), Float64(Float64(cbrt(Float64(x - -1.0)) * cbrt(x)) + (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], 0.0], N[(0.3333333333333333 * N[(N[(N[Power[x, -0.25], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[(N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $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)) < 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. 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 94.9%

      \[\leadsto \left(\frac{1}{{x}^{0.08333333333333333}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.7%

      \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{x}^{-0.25}}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]

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

   1. Initial program 64.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower--.f64 65.0

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

   4. Applied rewrites 65.0%

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

   6. Applied rewrites 97.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, x - -1\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{x - -1} \cdot \sqrt[3]{x} + {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x - -1} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 0.0)
   (* 0.3333333333333333 (* (/ (pow x -0.25) (cbrt x)) (pow (cbrt x) -0.25)))
   (/
    (- (- x -1.0) x)
    (fma
     (cbrt x)
     (+ (cbrt (- x -1.0)) (cbrt x))
     (pow (exp 0.6666666666666666) (log1p x))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 0.0) {
		tmp = 0.3333333333333333 * ((pow(x, -0.25) / cbrt(x)) * pow(cbrt(x), -0.25));
	} else {
		tmp = ((x - -1.0) - x) / fma(cbrt(x), (cbrt((x - -1.0)) + cbrt(x)), pow(exp(0.6666666666666666), log1p(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 0.0)
		tmp = Float64(0.3333333333333333 * Float64(Float64((x ^ -0.25) / cbrt(x)) * (cbrt(x) ^ -0.25)));
	else
		tmp = Float64(Float64(Float64(x - -1.0) - x) / fma(cbrt(x), Float64(cbrt(Float64(x - -1.0)) + cbrt(x)), (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], 0.0], N[(0.3333333333333333 * N[(N[(N[Power[x, -0.25], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - -1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $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)) < 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. 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 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 94.9%

      \[\leadsto \left(\frac{1}{{x}^{0.08333333333333333}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.7%

      \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{x}^{-0.25}}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]

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

   1. Initial program 64.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower--.f64 65.0

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

   4. Applied rewrites 65.0%

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

   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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -1\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x - -1} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.2e+77)
   (/
    (fma
     (cbrt (pow x 4.0))
     0.3333333333333333
     (fma
      (cbrt (/ 1.0 (* x x)))
      0.06172839506172839
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (* 0.3333333333333333 (* (/ (pow x -0.25) (cbrt x)) (pow (cbrt x) -0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.2e+77) {
		tmp = fma(cbrt(pow(x, 4.0)), 0.3333333333333333, fma(cbrt((1.0 / (x * x))), 0.06172839506172839, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = 0.3333333333333333 * ((pow(x, -0.25) / cbrt(x)) * pow(cbrt(x), -0.25));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.2e+77)
		tmp = Float64(fma(cbrt((x ^ 4.0)), 0.3333333333333333, fma(cbrt(Float64(1.0 / Float64(x * x))), 0.06172839506172839, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64((x ^ -0.25) / cbrt(x)) * (cbrt(x) ^ -0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.2e+77], N[(N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[Power[x, -0.25], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999999e77

   1. Initial program 21.6%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 24.6

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

   4. Applied rewrites 24.6%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   if 1.1999999999999999e77 < x

   1. Initial program 4.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 43.5

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 94.4%

      \[\leadsto \left(\frac{1}{{x}^{0.08333333333333333}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.7%

      \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.7%

       \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{x}^{-0.25}}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 27000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 27000000.0)
   (- (/ (cbrt (fma x x -1.0)) (cbrt (- x 1.0))) (cbrt x))
   (* 0.3333333333333333 (* (/ (pow x -0.25) (cbrt x)) (pow (cbrt x) -0.25)))))

C

double code(double x) {
	double tmp;
	if (x <= 27000000.0) {
		tmp = (cbrt(fma(x, x, -1.0)) / cbrt((x - 1.0))) - cbrt(x);
	} else {
		tmp = 0.3333333333333333 * ((pow(x, -0.25) / cbrt(x)) * pow(cbrt(x), -0.25));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 27000000.0)
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) / cbrt(Float64(x - 1.0))) - cbrt(x));
	else
		tmp = Float64(0.3333333333333333 * Float64(Float64((x ^ -0.25) / cbrt(x)) * (cbrt(x) ^ -0.25)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 27000000.0], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(x - 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(N[Power[x, -0.25], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], -0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e7

   1. Initial program 81.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower--.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if 2.7e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 94.4%

      \[\leadsto \left(\frac{1}{{x}^{0.08333333333333333}} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 98.0%

      \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{-1}}{{x}^{0.25}}\right) \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-0.25} \cdot \frac{{x}^{-0.25}}{\sqrt[3]{x}}\right) \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 27000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{{x}^{-0.25}}{\sqrt[3]{x}} \cdot {\left(\sqrt[3]{x}\right)}^{-0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 27000000:\\ \;\;\;\;\frac{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3 \cdot \sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 27000000.0)
   (- (/ (cbrt (fma x x -1.0)) (cbrt (- x 1.0))) (cbrt x))
   (/ (/ 1.0 (* 3.0 (cbrt x))) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 27000000.0) {
		tmp = (cbrt(fma(x, x, -1.0)) / cbrt((x - 1.0))) - cbrt(x);
	} else {
		tmp = (1.0 / (3.0 * cbrt(x))) / cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 27000000.0)
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) / cbrt(Float64(x - 1.0))) - cbrt(x));
	else
		tmp = Float64(Float64(1.0 / Float64(3.0 * cbrt(x))) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 27000000.0], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(x - 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(3.0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e7

   1. Initial program 81.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower--.f64 81.9

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

   4. Applied rewrites 81.9%

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

   if 2.7e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

Alternative 6: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{1}{\frac{-1}{\sqrt[3]{x} - \sqrt[3]{x - -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3 \cdot \sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (/ 1.0 (/ -1.0 (- (cbrt x) (cbrt (- x -1.0)))))
   (/ (/ 1.0 (* 3.0 (cbrt x))) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = 1.0 / (-1.0 / (cbrt(x) - cbrt((x - -1.0))));
	} else {
		tmp = (1.0 / (3.0 * cbrt(x))) / cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = 1.0 / (-1.0 / (Math.cbrt(x) - Math.cbrt((x - -1.0))));
	} else {
		tmp = (1.0 / (3.0 * Math.cbrt(x))) / Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(1.0 / Float64(-1.0 / Float64(cbrt(x) - cbrt(Float64(x - -1.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(3.0 * cbrt(x))) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(1.0 / N[(-1.0 / N[(N[Power[x, 1/3], $MachinePrecision] - N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(3.0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. cbrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-cbrt.f64 N/A

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

      12. lower--.f64 81.9

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

   4. Applied rewrites 81.9%

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

   6. Applied rewrites 81.1%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\frac{1}{\frac{-1}{\sqrt[3]{x} - \sqrt[3]{x - -1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3 \cdot \sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3 \cdot \sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (/ (/ 1.0 (* 3.0 (cbrt x))) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = (1.0 / (3.0 * cbrt(x))) / cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = (1.0 / (3.0 * Math.cbrt(x))) / Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(Float64(1.0 / Float64(3.0 * cbrt(x))) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(3.0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3 \cdot \sqrt[3]{x}}}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (/ (* (cbrt (/ 1.0 x)) 0.3333333333333333) (cbrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = (cbrt((1.0 / x)) * 0.3333333333333333) / cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = (Math.cbrt((1.0 / x)) * 0.3333333333333333) / Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(Float64(cbrt(Float64(1.0 / x)) * 0.3333333333333333) / cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (/ 1.0 (* (* 3.0 (cbrt x)) (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = 1.0 / ((3.0 * cbrt(x)) * cbrt(x));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = 1.0 / ((3.0 * Math.cbrt(x)) * Math.cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(1.0 / Float64(Float64(3.0 * cbrt(x)) * cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(3.0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (/ 1.0 (* (pow (cbrt x) 2.0) 3.0))))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = 1.0 / (pow(cbrt(x), 2.0) * 3.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = 1.0 / (Math.pow(Math.cbrt(x), 2.0) * 3.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64(1.0 / Float64((cbrt(x) ^ 2.0) * 3.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{2} \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 33000000.0)
   (- (cbrt (+ 1.0 x)) (cbrt x))
   (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = cbrt((1.0 + x)) - cbrt(x);
	} else {
		tmp = pow(cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 33000000.0) {
		tmp = Math.cbrt((1.0 + x)) - Math.cbrt(x);
	} else {
		tmp = Math.pow(Math.cbrt(x), -2.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 33000000.0)
		tmp = Float64(cbrt(Float64(1.0 + x)) - cbrt(x));
	else
		tmp = Float64((cbrt(x) ^ -2.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 33000000.0], N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3e7

   1. Initial program 81.0%

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

   if 3.3e7 < x

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf

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

      1. *-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 54.8

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

   5. Applied rewrites 54.8%

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

   7. Applied rewrites 97.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 33000000:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. 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 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 95.1%

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

Alternative 13: 92.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 12.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 92.7

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

   5. Applied rewrites 92.7%

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\frac{1}{x}}{x}} \cdot 0.3333333333333333} \]
   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 92.9%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-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 8.8

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

   5. Applied rewrites 8.8%

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

   7. Applied rewrites 89.1%

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

Alternative 14: 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 8.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. 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 54.0

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

5. Applied rewrites 54.0%

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

7. Applied rewrites 87.7%

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

Alternative 15: 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(x))
end

Wolfram

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

Derivation

1. Initial program 8.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. rem-cbrt-cube N/A

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

   5. sqr-pow N/A

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

   6. pow-prod-down N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg N/A

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

   10. pow-prod-down N/A

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

   11. sqr-pow N/A

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

   12. rem-cbrt-cube N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 5.5

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

7. Applied rewrites 5.5%

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

8. Final simplification 5.5%

   \[\leadsto 1 + \sqrt[3]{x} \]
9. Add Preprocessing

Alternative 16: 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 8.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024268 
(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)))