2cbrt (problem 3.3.4)

Percentage Accurate: 7.2% → 97.9%

Time: 10.0s

Alternatives: 15

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: 7.2% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\sqrt[3]{{\left({x}^{-0.25}\right)}^{4} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    (cbrt (+ (pow (pow x -0.25) 4.0) (/ 2.0 (* x x))))
    (+
     (cbrt (fma (pow x -0.5) (pow x -0.5) (/ 1.0 (* x x))))
     (cbrt (/ 1.0 x)))))))

C

double code(double x) {
	return 1.0 / (x * (cbrt((pow(pow(x, -0.25), 4.0) + (2.0 / (x * x)))) + (cbrt(fma(pow(x, -0.5), pow(x, -0.5), (1.0 / (x * x)))) + cbrt((1.0 / x)))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(cbrt(Float64(((x ^ -0.25) ^ 4.0) + Float64(2.0 / Float64(x * x)))) + Float64(cbrt(fma((x ^ -0.5), (x ^ -0.5), Float64(1.0 / Float64(x * x)))) + cbrt(Float64(1.0 / x))))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[Power[N[(N[Power[N[Power[x, -0.25], $MachinePrecision], 4.0], $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(N[Power[x, -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. rem-cube-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. sqr-pow N/A

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

   8. lower-fma.f64 N/A

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

9. Applied rewrites 98.5%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)}} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]
10. Step-by-step derivation

    1. inv-pow N/A

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

    2. metadata-eval N/A

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

    3. pow-pow N/A

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

    4. sqr-pow N/A

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

    5. pow2 N/A

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

    6. pow-pow N/A

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

    7. lower-pow.f64 N/A

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

    8. lower-pow.f64 N/A

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

    9. metadata-eval N/A

       \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{{\left({x}^{\color{blue}{\frac{-1}{4}}}\right)}^{\left(2 \cdot 2\right)} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{\frac{-1}{2}}, {x}^{\frac{-1}{2}}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]

    10. metadata-eval 98.5

        \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{{\left({x}^{-0.25}\right)}^{\color{blue}{4}} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]

11. Applied rewrites 98.5%

    \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\color{blue}{{\left({x}^{-0.25}\right)}^{4}} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]
12. Add Preprocessing

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right) + \sqrt[3]{\frac{2}{x \cdot x} + {x}^{-1}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    (+ (cbrt (fma (pow x -0.5) (pow x -0.5) (/ 1.0 (* x x)))) (cbrt (/ 1.0 x)))
    (cbrt (+ (/ 2.0 (* x x)) (pow x -1.0)))))))

C

double code(double x) {
	return 1.0 / (x * ((cbrt(fma(pow(x, -0.5), pow(x, -0.5), (1.0 / (x * x)))) + cbrt((1.0 / x))) + cbrt(((2.0 / (x * x)) + pow(x, -1.0)))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(Float64(cbrt(fma((x ^ -0.5), (x ^ -0.5), Float64(1.0 / Float64(x * x)))) + cbrt(Float64(1.0 / x))) + cbrt(Float64(Float64(2.0 / Float64(x * x)) + (x ^ -1.0))))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[(N[Power[N[(N[Power[x, -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[Power[x, -1.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. rem-cube-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. sqr-pow N/A

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

   8. lower-fma.f64 N/A

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

9. Applied rewrites 98.5%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)}} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]
10. Step-by-step derivation

    1. inv-pow N/A

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

    2. lower-pow.f64 98.5

       \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\color{blue}{{x}^{-1}} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]

11. Applied rewrites 98.5%

    \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\color{blue}{{x}^{-1}} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]

12. Final simplification 98.5%

    \[\leadsto \frac{1}{x \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right) + \sqrt[3]{\frac{2}{x \cdot x} + {x}^{-1}}\right)} \]
13. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right) + \sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    (+ (cbrt (fma (pow x -0.5) (pow x -0.5) (/ 1.0 (* x x)))) (cbrt (/ 1.0 x)))
    (cbrt (+ (/ 2.0 (* x x)) (/ 1.0 x)))))))

C

double code(double x) {
	return 1.0 / (x * ((cbrt(fma(pow(x, -0.5), pow(x, -0.5), (1.0 / (x * x)))) + cbrt((1.0 / x))) + cbrt(((2.0 / (x * x)) + (1.0 / x)))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(Float64(cbrt(fma((x ^ -0.5), (x ^ -0.5), Float64(1.0 / Float64(x * x)))) + cbrt(Float64(1.0 / x))) + cbrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))))))
end

Wolfram

code[x_] := N[(1.0 / N[(x * N[(N[(N[Power[N[(N[Power[x, -0.5], $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.9%

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. rem-cube-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. sqr-pow N/A

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

   8. lower-fma.f64 N/A

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

9. Applied rewrites 98.5%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \left(\sqrt[3]{\color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)}} + \sqrt[3]{\frac{1}{x}}\right)\right)} \]

10. Final simplification 98.5%

    \[\leadsto \frac{1}{x \cdot \left(\left(\sqrt[3]{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, \frac{1}{x \cdot x}\right)} + \sqrt[3]{\frac{1}{x}}\right) + \sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}}\right)} \]
11. Add Preprocessing

Alternative 4: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot \left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x \cdot x} + \frac{1}{x}}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (*
   x
   (+
    (cbrt (+ (/ 2.0 (* x x)) (/ 1.0 x)))
    (+ (cbrt (/ 1.0 x)) (cbrt (+ (/ 1.0 (* x x)) (/ 1.0 x))))))))

C

double code(double x) {
	return 1.0 / (x * (cbrt(((2.0 / (x * x)) + (1.0 / x))) + (cbrt((1.0 / x)) + cbrt(((1.0 / (x * x)) + (1.0 / x))))));
}

Java

public static double code(double x) {
	return 1.0 / (x * (Math.cbrt(((2.0 / (x * x)) + (1.0 / x))) + (Math.cbrt((1.0 / x)) + Math.cbrt(((1.0 / (x * x)) + (1.0 / x))))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(x * Float64(cbrt(Float64(Float64(2.0 / Float64(x * x)) + Float64(1.0 / x))) + Float64(cbrt(Float64(1.0 / x)) + cbrt(Float64(Float64(1.0 / Float64(x * x)) + Float64(1.0 / x)))))))
end

Wolfram

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

Derivation

1. Initial program 5.9%

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

   \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{2}{x \cdot x} + \frac{1}{x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x \cdot x} + \frac{1}{x}}\right)\right)} \]
9. Add Preprocessing

Alternative 5: 96.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

4. Applied rewrites 7.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cbrt.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in x around inf

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

   1. lower-/.f64 97.7

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

10. Applied rewrites 97.7%

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

11. Final simplification 97.7%

    \[\leadsto \frac{1}{x \cdot \left(\sqrt[3]{\frac{1}{x}} + \left(\sqrt[3]{\frac{1}{x}} + \sqrt[3]{\frac{1}{x \cdot x} + \frac{1}{x}}\right)\right)} \]
12. Add Preprocessing

Alternative 6: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.9%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.8

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

5. Applied rewrites 49.8%

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

   1. lift-*.f64 N/A

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

   2. cbrt-div N/A

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

   3. metadata-eval N/A

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

   4. inv-pow N/A

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

   5. pow1/3 N/A

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

   6. pow1/3 N/A

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

   7. lift-*.f64 N/A

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

   8. cbrt-prod N/A

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

   9. lift-cbrt.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. pow2 N/A

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

   12. pow-pow N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 97.3

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

7. Applied rewrites 97.3%

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

Alternative 7: 92.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg 96.5

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

      3. /-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. /-rgt-identity N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. frac-2neg N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval N/A

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

      20. frac-2neg N/A

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

      21. lower-/.f64 96.6

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

   9. Applied rewrites 96.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. pow-pow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. lower-*.f64 98.9

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

      19. lift-cbrt.f64 N/A

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

      20. pow1/3 N/A

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

      21. lift-/.f64 N/A

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

      22. inv-pow N/A

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

      23. pow-pow N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval 90.7

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

   9. Applied rewrites 90.7%

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

4. Final simplification 93.6%

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

Alternative 8: 92.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg 96.5

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

      3. /-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. /-rgt-identity N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. times-frac N/A

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

      10. *-rgt-identity N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 96.5

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

   9. Applied rewrites 96.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. pow-pow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. lower-*.f64 98.9

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

      19. lift-cbrt.f64 N/A

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

      20. pow1/3 N/A

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

      21. lift-/.f64 N/A

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

      22. inv-pow N/A

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

      23. pow-pow N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval 90.7

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

   9. Applied rewrites 90.7%

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

4. Final simplification 93.5%

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

Alternative 9: 92.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg 96.5

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

   9. Applied rewrites 96.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. *-rgt-identity N/A

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

      4. *-rgt-identity N/A

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

      5. pow1/3 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. pow-pow N/A

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

      14. inv-pow N/A

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

      15. lift-/.f64 N/A

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

      16. pow1/3 N/A

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

      17. lift-cbrt.f64 N/A

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

      18. lower-*.f64 98.9

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

      19. lift-cbrt.f64 N/A

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

      20. pow1/3 N/A

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

      21. lift-/.f64 N/A

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

      22. inv-pow N/A

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

      23. pow-pow N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval 90.7

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

   9. Applied rewrites 90.7%

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

4. Final simplification 93.5%

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

Alternative 10: 91.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg 96.5

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

   9. Applied rewrites 96.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. /-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. /-rgt-identity N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. cbrt-div N/A

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

      14. clear-num N/A

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

      15. cbrt-div N/A

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

      16. pow1/3 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-pow N/A

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

      20. pow1/3 N/A

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

      21. pow-div N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   9. Applied rewrites 89.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 91.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 7.1%

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

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-cbrt.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.8

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

   5. Applied rewrites 4.8%

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

      1. lift-*.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. cbrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cbrt.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 4.8

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

   7. Applied rewrites 4.8%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. remove-double-neg N/A

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

      4. /-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. /-rgt-identity N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. *-lft-identity N/A

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

      13. cbrt-div N/A

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

      14. clear-num N/A

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

      15. cbrt-div N/A

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

      16. pow1/3 N/A

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

      17. lift-/.f64 N/A

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

      18. inv-pow N/A

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

      19. pow-pow N/A

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

      20. pow1/3 N/A

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

      21. pow-div N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

   9. Applied rewrites 89.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\frac{1}{x}\right)}^{0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 88.7% 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 5.9%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.8

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

5. Applied rewrites 49.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. pow1/3 N/A

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

   4. lift-/.f64 N/A

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

   5. inv-pow N/A

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

   6. pow-pow N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. pow-pow N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow-flip N/A

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

   14. lift-pow.f64 N/A

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

   15. un-div-inv N/A

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

   16. lower-/.f64 89.4

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]

7. Applied rewrites 89.4%

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

Alternative 13: 88.7% accurate, 1.9× speedup

Math

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

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 5.9%

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

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-cbrt.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 49.8

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

5. Applied rewrites 49.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 49.8

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. pow-pow N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-pow.f64 N/A

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

   18. metadata-eval 89.4

       \[\leadsto {x}^{\color{blue}{-0.6666666666666666}} \cdot 0.3333333333333333 \]

7. Applied rewrites 89.4%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]

8. Final simplification 89.4%

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

Alternative 14: 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 5.9%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cbrt.f64 1.8

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

5. Applied rewrites 1.8%

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

Alternative 15: 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 5.9%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cbrt.f64 1.8

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

5. Applied rewrites 1.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \sqrt[3]{x}} \]
7. 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

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

   3. lower-cbrt.f64 1.8

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

8. Applied rewrites 1.8%

   \[\leadsto \color{blue}{-\sqrt[3]{x}} \]
9. 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 2024216 
(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)))