2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 98.9%

Time: 8.0s

Alternatives: 9

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: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 0.0)
		tmp = Float64((Float64(x / cbrt(x)) ^ -1.0) * 0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), exp(Float64(log1p(x) * 0.6666666666666666))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 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 54.7

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

   5. Applied rewrites 54.7%

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

   7. Applied rewrites 98.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

   9. Applied rewrites 99.0%

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

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

   1. Initial program 60.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 57.4

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

   4. Applied rewrites 57.4%

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

   6. Applied rewrites 97.9%

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

4. Final simplification 98.9%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{{\left(x \cdot x\right)}^{-1}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt[3]{x}}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+30)
   (/
    (fma
     (cbrt (pow x 4.0))
     0.3333333333333333
     (fma
      (cbrt (pow (* x x) -1.0))
      0.06172839506172839
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (* (pow (/ x (cbrt x)) -1.0) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+30) {
		tmp = fma(cbrt(pow(x, 4.0)), 0.3333333333333333, fma(cbrt(pow((x * x), -1.0)), 0.06172839506172839, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = pow((x / cbrt(x)), -1.0) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+30)
		tmp = Float64(fma(cbrt((x ^ 4.0)), 0.3333333333333333, fma(cbrt((Float64(x * x) ^ -1.0)), 0.06172839506172839, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64((Float64(x / cbrt(x)) ^ -1.0) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+30], N[(N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[Power[N[(x * x), $MachinePrecision], -1.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e30

   1. Initial program 28.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 95.5%

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

   if 2e30 < x

   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 51.4

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

   5. Applied rewrites 51.4%

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

   7. Applied rewrites 98.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

   9. Applied rewrites 99.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{{\left(x \cdot x\right)}^{-1}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt[3]{x}}\right)}^{-1} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 94.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 98.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 89.1%

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

4. Final simplification 92.2%

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

Alternative 4: 92.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.2%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.7

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

   5. Applied rewrites 94.7%

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

   7. Applied rewrites 94.8%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.9%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 6.9

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

   5. Applied rewrites 6.9%

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

   7. Applied rewrites 98.4%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]
   8. Step-by-step derivation

   9. Applied rewrites 89.1%

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

4. Final simplification 92.2%

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

Alternative 5: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{x}{\sqrt[3]{x}}\right)}^{-1} \cdot 0.3333333333333333 \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.2%

   \[\leadsto \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \]

9. Applied rewrites 96.8%

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

10. Final simplification 96.8%

    \[\leadsto {\left(\frac{x}{\sqrt[3]{x}}\right)}^{-1} \cdot 0.3333333333333333 \]
11. Add Preprocessing

Alternative 6: 89.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 96.3%

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

9. Applied rewrites 88.7%

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

Alternative 7: 89.0% 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 7.3%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.9

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

5. Applied rewrites 54.9%

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

7. Applied rewrites 88.7%

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

Alternative 8: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 7.3%

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

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. inv-pow N/A

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

   7. metadata-eval N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 5.3%

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

5. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-cbrt.f64 1.8

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

7. Applied rewrites 1.8%

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

9. Applied rewrites 1.8%

   \[\leadsto -{x}^{0.3333333333333333} \]
10. Step-by-step derivation

11. Applied rewrites 5.5%

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

Alternative 9: 4.2% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.3%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 8.4

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

4. Applied rewrites 8.4%

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

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

   11. pow1/3 N/A

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

   12. metadata-eval N/A

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

   13. pow-pow N/A

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

   14. inv-pow N/A

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. rem-exp-log N/A

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

   18. neg-log N/A

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

   19. lift-log1p.f64 N/A

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

   20. lift-neg.f64 N/A

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

   21. lift-exp.f64 N/A

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

   22. lift-pow.f64 N/A

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

6. Applied rewrites 8.4%

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

7. Taylor expanded in x around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

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

   4. mul0-rgt 4.1

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

9. Applied rewrites 4.1%

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

Developer Target 1: 98.4% 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 2024312 
(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)))