2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 99.0%

Time: 7.7s

Alternatives: 13

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.0% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1 + x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, t\_0 \cdot \sqrt[3]{x} + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 5e-11)
     (/ 0.3333333333333333 (/ x (cbrt x)))
     (/
      (fma -1.0 x (+ 1.0 x))
      (fma
       (cbrt x)
       (cbrt x)
       (+ (* t_0 (cbrt x)) (exp (* 0.6666666666666666 (log1p x)))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	} else {
		tmp = fma(-1.0, x, (1.0 + x)) / fma(cbrt(x), cbrt(x), ((t_0 * cbrt(x)) + exp((0.6666666666666666 * log1p(x)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	else
		tmp = Float64(fma(-1.0, x, Float64(1.0 + x)) / fma(cbrt(x), cbrt(x), Float64(Float64(t_0 * cbrt(x)) + exp(Float64(0.6666666666666666 * log1p(x))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[(N[(t$95$0 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.2%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 63.1%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 63.7

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

   4. Applied rewrites 63.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. rem-square-sqrt N/A

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

      7. +-commutative N/A

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

      8. rem-exp-log N/A

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

      9. lift-log1p.f64 N/A

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

      10. lift-exp.f64 N/A

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

      11. +-commutative N/A

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

      12. flip3-+ N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 98.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1 + x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{x} + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({\left(\sqrt[3]{x}\right)}^{2} + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 5e-11)
   (/ 0.3333333333333333 (/ x (cbrt x)))
   (/
    (- (+ 1.0 x) x)
    (+
     (cbrt (fma x x x))
     (+ (pow (cbrt x) 2.0) (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	} else {
		tmp = ((1.0 + x) - x) / (cbrt(fma(x, x, x)) + (pow(cbrt(x), 2.0) + exp((0.6666666666666666 * log1p(x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(cbrt(fma(x, x, x)) + Float64((cbrt(x) ^ 2.0) + exp(Float64(0.6666666666666666 * log1p(x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.2%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 63.1%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 61.5

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

   4. Applied rewrites 61.5%

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

   6. Applied rewrites 98.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + \left({\left(\sqrt[3]{x}\right)}^{2} + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(\sqrt[3]{x}\right)}^{2}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (cbrt (+ 1.0 x)) (cbrt x)) 5e-11)
   (/ 0.3333333333333333 (/ x (cbrt x)))
   (/
    (- (+ 1.0 x) x)
    (+
     (+ (cbrt (fma x x x)) (pow (cbrt x) 2.0))
     (exp (* 0.6666666666666666 (log1p x)))))))

C

double code(double x) {
	double tmp;
	if ((cbrt((1.0 + x)) - cbrt(x)) <= 5e-11) {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	} else {
		tmp = ((1.0 + x) - x) / ((cbrt(fma(x, x, x)) + pow(cbrt(x), 2.0)) + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(cbrt(Float64(1.0 + x)) - cbrt(x)) <= 5e-11)
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(Float64(cbrt(fma(x, x, x)) + (cbrt(x) ^ 2.0)) + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 5e-11], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(N[Power[N[(x * x + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x)) < 5.00000000000000018e-11

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.2%

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

   if 5.00000000000000018e-11 < (-.f64 (cbrt.f64 (+.f64 x #s(literal 1 binary64))) (cbrt.f64 x))

   1. Initial program 63.1%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 63.7

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

   4. Applied rewrites 63.7%

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

   6. Applied rewrites 98.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\left(\sqrt[3]{\mathsf{fma}\left(x, x, x\right)} + {\left(\sqrt[3]{x}\right)}^{2}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+14)
   (/
    (- (+ 1.0 x) x)
    (fma
     (cbrt x)
     (+ (cbrt (+ 1.0 x)) (cbrt x))
     (exp (* 0.6666666666666666 (log1p x)))))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+14) {
		tmp = ((1.0 + x) - x) / fma(cbrt(x), (cbrt((1.0 + x)) + cbrt(x)), exp((0.6666666666666666 * log1p(x))));
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(cbrt(x), Float64(cbrt(Float64(1.0 + x)) + cbrt(x)), exp(Float64(0.6666666666666666 * log1p(x)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+14], 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[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e14

   1. Initial program 63.1%

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

      1. lift-+.f64 N/A

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

      2. rem-cube-cbrt N/A

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

      3. lift-cbrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. unpow1/2 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow-pow N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 63.7

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

   4. Applied rewrites 63.7%

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

   6. Applied rewrites 98.0%

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

   if 5e14 < 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 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.06172839506172839, -0.1111111111111111 \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x}{\sqrt[3]{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5e+15)
   (/
    (fma
     (cbrt (pow x 4.0))
     0.3333333333333333
     (fma
      (cbrt (/ 1.0 (* x x)))
      0.06172839506172839
      (* -0.1111111111111111 (cbrt x))))
    (* x x))
   (/ 0.3333333333333333 (/ x (cbrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5e+15) {
		tmp = fma(cbrt(pow(x, 4.0)), 0.3333333333333333, fma(cbrt((1.0 / (x * x))), 0.06172839506172839, (-0.1111111111111111 * cbrt(x)))) / (x * x);
	} else {
		tmp = 0.3333333333333333 / (x / cbrt(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5e+15)
		tmp = Float64(fma(cbrt((x ^ 4.0)), 0.3333333333333333, fma(cbrt(Float64(1.0 / Float64(x * x))), 0.06172839506172839, Float64(-0.1111111111111111 * cbrt(x)))) / Float64(x * x));
	else
		tmp = Float64(0.3333333333333333 / Float64(x / cbrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5e+15], N[(N[(N[Power[N[Power[x, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.06172839506172839 + N[(-0.1111111111111111 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(x / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e15

   1. Initial program 63.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. lower-cbrt.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 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. 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 83.1%

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

   if 1.5e15 < 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 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 98.4%

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

   9. Applied rewrites 99.1%

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

   11. Applied rewrites 99.2%

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

Alternative 6: 97.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 96.3%

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

11. Applied rewrites 96.4%

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

Alternative 7: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 95.8%

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

9. Applied rewrites 96.4%

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

11. Applied rewrites 96.4%

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

Alternative 8: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 95.8%

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

9. Applied rewrites 96.4%

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

11. Applied rewrites 96.4%

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

Alternative 9: 97.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 95.8%

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

9. Applied rewrites 95.6%

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

11. Applied rewrites 96.3%

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

Alternative 10: 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 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 95.7%

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

9. Applied rewrites 88.2%

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

Alternative 11: 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 7.9%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 50.7

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

5. Applied rewrites 50.7%

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

7. Applied rewrites 88.2%

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

Alternative 12: 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 7.9%

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

Alternative 13: 1.8% 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 Float64(-cbrt(x))
end

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. lift-cbrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. clear-num N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. rem-cube-cbrt N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow-to-exp N/A

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

   15. rec-exp N/A

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

   16. rem-log-exp N/A

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

   17. pow-to-exp N/A

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

4. Applied rewrites 7.3%

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

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. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x + 1}\\ \frac{1}{\left(t\_0 \cdot t\_0 + \sqrt[3]{x} \cdot t\_0\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ x 1.0))))
   (/ 1.0 (+ (+ (* t_0 t_0) (* (cbrt x) t_0)) (* (cbrt x) (cbrt x))))))

C

double code(double x) {
	double t_0 = cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (cbrt(x) * t_0)) + (cbrt(x) * cbrt(x)));
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((x + 1.0));
	return 1.0 / (((t_0 * t_0) + (Math.cbrt(x) * t_0)) + (Math.cbrt(x) * Math.cbrt(x)));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x + 1.0))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_0) + Float64(cbrt(x) * t_0)) + Float64(cbrt(x) * cbrt(x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024332 
(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)))