2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.5%

Time: 7.8s

Alternatives: 11

Speedup: 1.9×

Specification

\[x > 1 \land x < 10^{+308}\]

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\\ t_1 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\ t_2 := t\_1 \cdot \sqrt[3]{x}\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{+118}:\\ \;\;\;\;{\left(\mathsf{fma}\left({t\_1}^{3}, x, {\left(1 + x\right)}^{2}\right)\right)}^{-1} \cdot \mathsf{fma}\left(t\_0, t\_0 - t\_2, {t\_2}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{x}} \cdot 0.3333333333333333}{\sqrt[3]{-x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp 0.6666666666666666) (log1p x)))
        (t_1 (+ (cbrt (+ 1.0 x)) (cbrt x)))
        (t_2 (* t_1 (cbrt x))))
   (if (<= x 3.2e+118)
     (*
      (pow (fma (pow t_1 3.0) x (pow (+ 1.0 x) 2.0)) -1.0)
      (fma t_0 (- t_0 t_2) (pow t_2 2.0)))
     (/ (* (cbrt (/ -1.0 x)) 0.3333333333333333) (cbrt (- x))))))

C

double code(double x) {
	double t_0 = pow(exp(0.6666666666666666), log1p(x));
	double t_1 = cbrt((1.0 + x)) + cbrt(x);
	double t_2 = t_1 * cbrt(x);
	double tmp;
	if (x <= 3.2e+118) {
		tmp = pow(fma(pow(t_1, 3.0), x, pow((1.0 + x), 2.0)), -1.0) * fma(t_0, (t_0 - t_2), pow(t_2, 2.0));
	} else {
		tmp = (cbrt((-1.0 / x)) * 0.3333333333333333) / cbrt(-x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(0.6666666666666666) ^ log1p(x)
	t_1 = Float64(cbrt(Float64(1.0 + x)) + cbrt(x))
	t_2 = Float64(t_1 * cbrt(x))
	tmp = 0.0
	if (x <= 3.2e+118)
		tmp = Float64((fma((t_1 ^ 3.0), x, (Float64(1.0 + x) ^ 2.0)) ^ -1.0) * fma(t_0, Float64(t_0 - t_2), (t_2 ^ 2.0)));
	else
		tmp = Float64(Float64(cbrt(Float64(-1.0 / x)) * 0.3333333333333333) / cbrt(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[0.6666666666666666], $MachinePrecision], N[Log[1 + x], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.2e+118], N[(N[Power[N[(N[Power[t$95$1, 3.0], $MachinePrecision] * x + N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(t$95$0 * N[(t$95$0 - t$95$2), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $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 < 3.20000000000000016e118

   1. Initial program 10.1%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-log1p.f64 9.2

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

   4. Applied rewrites 9.2%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. exp-to-pow N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. flip3-- N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-cbrt.f64 N/A

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

      15. rem-cube-cbrt N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

   6. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]

   8. Applied rewrites 98.4%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left({\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}^{3}, x, {\left(1 + x\right)}^{2}\right)\right)}^{-1} \cdot \mathsf{fma}\left({\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} - \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}, {\left(\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}^{2}\right)} \]

   if 3.20000000000000016e118 < x

   1. Initial program 4.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 28.6

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

   5. Applied rewrites 28.6%

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

   7. Applied rewrites 98.4%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{x + 1} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ x 1.0)) (cbrt x)) 0.0)
   (* (/ (cbrt (/ -1.0 x)) (cbrt (- x))) 0.3333333333333333)
   (/
    (- (+ 1.0 x) x)
    (fma (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)) (cbrt (pow (+ 1.0 x) 2.0))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((x + 1.0)) - cbrt(x)) <= 0.0) {
		tmp = (cbrt((-1.0 / x)) / cbrt(-x)) * 0.3333333333333333;
	} else {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), cbrt(pow((1.0 + x), 2.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(x + 1.0)) - cbrt(x)) <= 0.0)
		tmp = Float64(Float64(cbrt(Float64(-1.0 / x)) / cbrt(Float64(-x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), cbrt((Float64(1.0 + x) ^ 2.0))));
	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[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[(-x), 1/3], $MachinePrecision]), $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[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $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. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. 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 98.4%

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

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

   1. Initial program 56.2%

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

      1. lift-cbrt.f64 N/A

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

      2. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-log1p.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. exp-to-pow N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. flip3-- N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-cbrt.f64 N/A

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

      15. rem-cube-cbrt N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

   6. Applied rewrites 97.2%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube N/A

         \[\leadsto \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{\sqrt[3]{{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{3}}\right)}^{3}}}\right)} \]

      2. lift-exp.f64 N/A

         \[\leadsto \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\color{blue}{\left(e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{3}}\right)}}^{3}}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{2}{3}}}\right)}^{3}}\right)} \]

      4. lift-log1p.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. exp-to-pow N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. lower-cbrt.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 98.4

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

   8. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+15)
   (pow
    (fma
     (cbrt x)
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (pow (+ 1.0 x) 0.6666666666666666))
    -1.0)
   (* (/ (cbrt (/ -1.0 x)) (cbrt (- x))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1e+15) {
		tmp = pow(fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), pow((1.0 + x), 0.6666666666666666)), -1.0);
	} else {
		tmp = (cbrt((-1.0 / x)) / cbrt(-x)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+15)
		tmp = fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), (Float64(1.0 + x) ^ 0.6666666666666666)) ^ -1.0;
	else
		tmp = Float64(Float64(cbrt(Float64(-1.0 / x)) / cbrt(Float64(-x))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+15], N[Power[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[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[Power[N[(-1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[(-x), 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e15

   1. Initial program 63.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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. pow1/3 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. rem-cube-cbrt N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-log1p.f64 59.1

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

   4. Applied rewrites 59.1%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

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

      5. exp-to-pow N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. pow1/3 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. flip3-- N/A

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

      11. lower-/.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. rem-cube-cbrt N/A

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

      14. lift-cbrt.f64 N/A

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

      15. rem-cube-cbrt N/A

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

      16. lower--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

   6. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}\right)}} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{e^{\mathsf{log1p}\left(x\right) \cdot \frac{2}{3}}}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \frac{2}{3}}}\right)} \]

      3. lift-log1p.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 97.7

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

   8. Applied rewrites 97.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval 98.3

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

   10. Applied rewrites 98.3%

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

   if 1e15 < 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. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 55.2

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

   5. Applied rewrites 55.2%

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

   7. Applied rewrites 98.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+15}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \cdot 0.3333333333333333 \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 55.2

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

5. Applied rewrites 55.2%

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

7. Applied rewrites 96.5%

   \[\leadsto \frac{\sqrt[3]{\frac{-1}{x}}}{\sqrt[3]{-x}} \cdot 0.3333333333333333 \]
8. Add Preprocessing

Alternative 5: 96.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 0.3333333333333333 (pow (cbrt x) 2.0)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 55.2

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

5. Applied rewrites 55.2%

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

7. Applied rewrites 96.5%

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

Alternative 6: 96.5% 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 6.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 55.2

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

5. Applied rewrites 55.2%

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

7. Applied rewrites 96.5%

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

Alternative 7: 92.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 95.3

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

   5. Applied rewrites 95.3%

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

   7. Applied rewrites 94.9%

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

   9. Applied rewrites 95.5%

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

   if 1.35000000000000003e154 < 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. unpow2 N/A

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

      4. sqr-neg-rev N/A

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

      5. associate-/r* N/A

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

      6. distribute-neg-frac2 N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. lower-cbrt.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. associate-/r* N/A

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

      14. sqr-neg-rev N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 7.5

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

   5. Applied rewrites 7.5%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 89.1%

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

Alternative 8: 88.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 55.2

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

5. Applied rewrites 55.2%

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 88.9%

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

Alternative 9: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. sqr-neg-rev N/A

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

   5. associate-/r* N/A

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

   6. distribute-neg-frac2 N/A

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

   7. distribute-neg-frac N/A

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

   8. metadata-eval N/A

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

   9. lower-cbrt.f64 N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. distribute-neg-frac2 N/A

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

   13. associate-/r* N/A

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

   14. sqr-neg-rev N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lower-/.f64 55.2

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

5. Applied rewrites 55.2%

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

7. Applied rewrites 88.9%

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

Alternative 10: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

   1. lift-cbrt.f64 N/A

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

   2. pow1/3 N/A

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

   3. lower-pow.f64 1.8

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

7. Applied rewrites 1.8%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. sqr-neg-rev N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. unpow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow1/3 N/A

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

   10. lift-cbrt.f64 5.4

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

9. Applied rewrites 5.4%

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

Alternative 11: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

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

Developer Target 1: 98.4% 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 2024337 
(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)))