2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.4%

Time: 3.3s

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({x}^{-1.6666666666666667}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, {x}^{-0.6666666666666666} \cdot -0.1111111111111111\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (pow x -1.6666666666666667)
   0.06172839506172839
   (fma
    0.3333333333333333
    (cbrt x)
    (* (pow x -0.6666666666666666) -0.1111111111111111)))
  x))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), 0.06172839506172839, fma(0.3333333333333333, cbrt(x), (pow(x, -0.6666666666666666) * -0.1111111111111111))) / x;
}

Julia

function code(x)
	return Float64(fma((x ^ -1.6666666666666667), 0.06172839506172839, fma(0.3333333333333333, cbrt(x), Float64((x ^ -0.6666666666666666) * -0.1111111111111111))) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * 0.06172839506172839 + N[(0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision] + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   8. lower-pow.f64 98.4

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x} \]

6. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x}} \]

7. Taylor expanded in x around -inf

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   6. lower-cbrt.f64 N/A

      \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   9. lower-pow.f64 98.4

      \[\leadsto \frac{0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}} - \mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

9. Applied rewrites 98.4%

   \[\leadsto \frac{0.06172839506172839 \cdot \frac{1}{{x}^{1.6666666666666667}} - \mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

       \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

    2. lift-cbrt.f64 N/A

       \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} - \mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

    3. lift-fma.f64 N/A

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

    4. associate--r+ N/A

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

    5. sub-flip N/A

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

    6. fp-cancel-sub-sign-inv N/A

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

    7. metadata-eval N/A

       \[\leadsto \frac{\left(\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right) + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)}{x} \]

    8. associate-+l+ N/A

       \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \left(\frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    9. lift-*.f64 N/A

       \[\leadsto \frac{\frac{5}{81} \cdot \frac{1}{{x}^{\frac{5}{3}}} + \left(\frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    10. *-commutative N/A

        \[\leadsto \frac{\frac{1}{{x}^{\frac{5}{3}}} \cdot \frac{5}{81} + \left(\frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    11. lower-fma.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{{x}^{\frac{5}{3}}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    12. lift-/.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{{x}^{\frac{5}{3}}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    13. lift-pow.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{{x}^{\frac{5}{3}}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    14. pow-flip N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    15. lower-pow.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(\mathsf{neg}\left(\frac{5}{3}\right)\right)}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    16. metadata-eval N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{5}{81}, \frac{1}{3} \cdot \sqrt[3]{x} + \left(\mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

    17. lower-fma.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{-5}{3}}, \frac{5}{81}, \mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \mathsf{neg}\left(\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)\right)}{x} \]

11. Applied rewrites 98.4%

    \[\leadsto \frac{\mathsf{fma}\left({x}^{-1.6666666666666667}, 0.06172839506172839, \mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, {x}^{-0.6666666666666666} \cdot -0.1111111111111111\right)\right)}{x} \]
12. Add Preprocessing

Alternative 2: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \left(--0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(-\frac{{x}^{-0.6666666666666666} \cdot 0.1111111111111111}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (- (* -0.3333333333333333 (/ (cbrt x) x)))
  (- (/ (* (pow x -0.6666666666666666) 0.1111111111111111) x))))

C

double code(double x) {
	return -(-0.3333333333333333 * (cbrt(x) / x)) + -((pow(x, -0.6666666666666666) * 0.1111111111111111) / x);
}

Java

public static double code(double x) {
	return -(-0.3333333333333333 * (Math.cbrt(x) / x)) + -((Math.pow(x, -0.6666666666666666) * 0.1111111111111111) / x);
}

Julia

function code(x)
	return Float64(Float64(-Float64(-0.3333333333333333 * Float64(cbrt(x) / x))) + Float64(-Float64(Float64((x ^ -0.6666666666666666) * 0.1111111111111111) / x)))
end

Wolfram

code[x_] := N[((-N[(-0.3333333333333333 * N[(N[Power[x, 1/3], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]) + (-N[(N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{\color{blue}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   6. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   7. lower-pow.f64 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

6. Applied rewrites 98.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   4. lift-cbrt.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   5. lift-fma.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}\right) \]

   6. div-add N/A

      \[\leadsto \mathsf{neg}\left(\left(\frac{\frac{-1}{3} \cdot \sqrt[3]{x}}{x} + \frac{\frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}\right)\right) \]

   7. distribute-neg-in N/A

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

   8. lower-+.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-/l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-/.f64 98.1

       \[\leadsto \left(--0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\right) + \left(-\frac{0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}}{x}\right) \]

8. Applied rewrites 98.1%

   \[\leadsto \left(--0.3333333333333333 \cdot \frac{\sqrt[3]{x}}{x}\right) + \color{blue}{\left(-\frac{{x}^{-0.6666666666666666} \cdot 0.1111111111111111}{x}\right)} \]
9. Add Preprocessing

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, {x}^{-0.6666666666666666} \cdot 0.1111111111111111\right)}{-x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   -0.3333333333333333
   (cbrt x)
   (* (pow x -0.6666666666666666) 0.1111111111111111))
  (- x)))

C

double code(double x) {
	return fma(-0.3333333333333333, cbrt(x), (pow(x, -0.6666666666666666) * 0.1111111111111111)) / -x;
}

Julia

function code(x)
	return Float64(fma(-0.3333333333333333, cbrt(x), Float64((x ^ -0.6666666666666666) * 0.1111111111111111)) / Float64(-x))
end

Wolfram

code[x_] := N[(N[(-0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision] + N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * 0.1111111111111111), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{\color{blue}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   6. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   7. lower-pow.f64 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

6. Applied rewrites 98.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   4. distribute-neg-frac2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{\mathsf{neg}\left(x\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   9. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   11. pow-flip N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, {x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   12. lower-pow.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, {x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, {x}^{\frac{-2}{3}} \cdot \frac{1}{9}\right)}{\mathsf{neg}\left(x\right)} \]

   14. lower-neg.f64 98.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, {x}^{-0.6666666666666666} \cdot 0.1111111111111111\right)}{-x} \]

8. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, {x}^{-0.6666666666666666} \cdot 0.1111111111111111\right)}{\color{blue}{-x}} \]
9. Add Preprocessing

Alternative 4: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{-0.6666666666666666} \cdot -0.1111111111111111 - -0.3333333333333333 \cdot \sqrt[3]{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (* (pow x -0.6666666666666666) -0.1111111111111111)
   (* -0.3333333333333333 (cbrt x)))
  x))

C

double code(double x) {
	return ((pow(x, -0.6666666666666666) * -0.1111111111111111) - (-0.3333333333333333 * cbrt(x))) / x;
}

Java

public static double code(double x) {
	return ((Math.pow(x, -0.6666666666666666) * -0.1111111111111111) - (-0.3333333333333333 * Math.cbrt(x))) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64((x ^ -0.6666666666666666) * -0.1111111111111111) - Float64(-0.3333333333333333 * cbrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -0.6666666666666666], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] - N[(-0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{x}} \]

   2. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\frac{-1}{3} \cdot \sqrt[3]{x} + \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}}{\color{blue}{x}} \]

   3. lower-fma.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   5. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   6. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x} \]

   7. lower-pow.f64 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

6. Applied rewrites 98.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x}} \]
7. Step-by-step derivation

   1. sub-to-mult-rev 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

   2. metadata-eval 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

   3. add-flip 98.1

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \sqrt[3]{x}, 0.1111111111111111 \cdot \frac{1}{{x}^{0.6666666666666666}}\right)}{x} \]

   4. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}} \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)}{x}\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)}{\color{blue}{x}} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(\frac{-1}{3}, \sqrt[3]{x}, \frac{1}{9} \cdot \frac{1}{{x}^{\frac{2}{3}}}\right)\right)}{\color{blue}{x}} \]

8. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{{x}^{-0.6666666666666666} \cdot -0.1111111111111111 - -0.3333333333333333 \cdot \sqrt[3]{x}}{x}} \]
9. Add Preprocessing

Alternative 5: 96.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (/ 1.0 (* (cbrt x) 3.0)) (cbrt x)))

C

double code(double x) {
	return (1.0 / (cbrt(x) * 3.0)) / cbrt(x);
}

Java

public static double code(double x) {
	return (1.0 / (Math.cbrt(x) * 3.0)) / Math.cbrt(x);
}

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(cbrt(x) * 3.0)) / cbrt(x))
end

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 96.5

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

6. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-/.f64 N/A

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

   4. mult-flip N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 96.5

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

8. Applied rewrites 96.5%

   \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot 3}}{\sqrt[3]{\color{blue}{x}}} \]
9. Add Preprocessing

Alternative 6: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 96.5

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

6. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. inv-pow N/A

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

   6. lower-*.f64 N/A

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

   7. inv-pow N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg-rev N/A

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

   10. lower-/.f64 96.4

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

8. Applied rewrites 96.4%

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

Alternative 7: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. pow1/3 N/A

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

   5. lift-cbrt.f64 N/A

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

   6. pow1/3 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 96.5

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

6. Applied rewrites 96.5%

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

Alternative 8: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. div-flip N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow1/3 N/A

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

   15. pow1/3 N/A

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

   16. pow-sqr N/A

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

   17. metadata-eval N/A

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

   18. pow-flip N/A

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

   19. lower-pow.f64 N/A

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

   20. metadata-eval 88.8

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

6. Applied rewrites 88.8%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. pow-cbrt N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow2 N/A

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

   8. pow2 N/A

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

   9. pow-flip N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval 96.5

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

8. Applied rewrites 96.5%

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

Alternative 9: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+231}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{1}{-\sqrt{\sqrt[3]{x} \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{e^{\log x \cdot 0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6e+231)
   (* -0.3333333333333333 (/ 1.0 (- (sqrt (* (cbrt x) x)))))
   (/ 0.3333333333333333 (exp (* (log x) 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6e+231) {
		tmp = -0.3333333333333333 * (1.0 / -sqrt((cbrt(x) * x)));
	} else {
		tmp = 0.3333333333333333 / exp((log(x) * 0.6666666666666666));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6e+231) {
		tmp = -0.3333333333333333 * (1.0 / -Math.sqrt((Math.cbrt(x) * x)));
	} else {
		tmp = 0.3333333333333333 / Math.exp((Math.log(x) * 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6e+231)
		tmp = Float64(-0.3333333333333333 * Float64(1.0 / Float64(-sqrt(Float64(cbrt(x) * x)))));
	else
		tmp = Float64(0.3333333333333333 / exp(Float64(log(x) * 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.6e+231], N[(-0.3333333333333333 * N[(1.0 / (-N[Sqrt[N[(N[Power[x, 1/3], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Exp[N[(N[Log[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000016e231

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. pow1/3 N/A

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

      11. lift-cbrt.f64 N/A

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

      12. pow1/3 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lift-cbrt.f64 N/A

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

      16. lift-cbrt.f64 N/A

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

      17. pow1/3 N/A

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

      18. pow1/3 N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. lift-pow.f64 88.8

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{-{x}^{0.6666666666666666}} \]

   6. Applied rewrites 88.8%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{-{x}^{0.6666666666666666}}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-e^{\log x \cdot \frac{2}{3}}} \]

      3. exp-fabs N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-\left|e^{\log x \cdot \frac{2}{3}}\right|} \]

      4. pow-to-exp N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-\left|{x}^{\frac{2}{3}}\right|} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-\left|{x}^{\frac{2}{3}}\right|} \]

      6. rem-sqrt-square-rev N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 67.8

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{-\sqrt{{x}^{1.3333333333333333}}} \]

   8. Applied rewrites 67.8%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{-\sqrt{{x}^{1.3333333333333333}}} \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-\sqrt{{x}^{\frac{4}{3}}}} \]

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. metadata-eval N/A

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

      5. pow-cbrt N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-cbrt N/A

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

      12. unpow2 N/A

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

      13. cube-mult N/A

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

      14. rem-cube-cbrt N/A

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

      15. lower-*.f64 N/A

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

      16. lift-cbrt.f64 73.5

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

   10. Applied rewrites 73.5%

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

   if 1.60000000000000016e231 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto \frac{\frac{1}{3}}{e^{\log x \cdot \frac{2}{3}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{e^{\log x \cdot \frac{2}{3}}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{e^{\log x \cdot \frac{2}{3}}} \]

      5. lower-log.f64 89.2

         \[\leadsto \frac{0.3333333333333333}{e^{\log x \cdot 0.6666666666666666}} \]

   6. Applied rewrites 89.2%

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

Alternative 10: 92.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 49.9

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

   6. Applied rewrites 49.9%

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

   if 1.31999999999999998e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-sqr N/A

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

      10. pow1/3 N/A

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

      11. lift-cbrt.f64 N/A

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

      12. pow1/3 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lift-cbrt.f64 N/A

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

      16. lift-cbrt.f64 N/A

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

      17. pow1/3 N/A

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

      18. pow1/3 N/A

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

      19. pow-sqr N/A

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

      20. metadata-eval N/A

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

      21. lift-pow.f64 88.8

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{-{x}^{0.6666666666666666}} \]

   6. Applied rewrites 88.8%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{1}{-{x}^{0.6666666666666666}}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-e^{\log x \cdot \frac{2}{3}}} \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{-1}{3} \cdot \frac{1}{-e^{\log x \cdot \frac{2}{3}}} \]

      4. rem-cube-cbrt N/A

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

      5. lift-cbrt.f64 N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-*.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. lift-cbrt.f64 N/A

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

      12. rem-cube-cbrt N/A

          \[\leadsto \frac{-1}{3} \cdot \frac{1}{-e^{\log x \cdot \frac{2}{3}}} \]

      13. lower-log.f64 89.2

          \[\leadsto -0.3333333333333333 \cdot \frac{1}{-e^{\log x \cdot 0.6666666666666666}} \]

   8. Applied rewrites 89.2%

      \[\leadsto -0.3333333333333333 \cdot \frac{1}{-e^{\log x \cdot 0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 92.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 49.9

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

   6. Applied rewrites 49.9%

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

   if 1.31999999999999998e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. div-flip N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow1/3 N/A

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

      16. pow-sqr N/A

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

      17. metadata-eval N/A

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

      18. pow-flip N/A

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

      19. lower-pow.f64 N/A

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

      20. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

      \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      3. lower-exp.f64 N/A

         \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      4. rem-cube-cbrt N/A

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

      5. lift-cbrt.f64 N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

      8. lower-*.f64 N/A

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

      9. rem-log-exp N/A

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

      10. pow-to-exp N/A

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

      11. lift-cbrt.f64 N/A

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

      12. rem-cube-cbrt N/A

          \[\leadsto e^{\log x \cdot \frac{-2}{3}} \cdot \frac{1}{3} \]

      13. lower-log.f64 89.2

          \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]

   8. Applied rewrites 89.2%

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

Alternative 12: 92.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.31999999999999998e154

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. pow1/3 N/A

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

      5. pow1/3 N/A

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

      6. cbrt-unprod N/A

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

      7. lower-cbrt.f64 N/A

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

      8. lower-*.f64 49.9

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

   6. Applied rewrites 49.9%

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

   if 1.31999999999999998e154 < x

   1. Initial program 7.0%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow1/3 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. pow1/3 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. div-flip N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lift-cbrt.f64 N/A

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

      13. lift-cbrt.f64 N/A

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

      14. pow1/3 N/A

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

      15. pow1/3 N/A

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

      16. pow-sqr N/A

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

      17. metadata-eval N/A

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

      18. pow-flip N/A

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

      19. lower-pow.f64 N/A

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

      20. metadata-eval 88.8

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

   6. Applied rewrites 88.8%

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

Alternative 13: 88.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

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

Alternative 14: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.0%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.8

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

4. Applied rewrites 88.8%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{x}^{0.6666666666666666}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. pow1/3 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. pow1/3 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. div-flip N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. lift-cbrt.f64 N/A

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

   14. pow1/3 N/A

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

   15. pow1/3 N/A

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

   16. pow-sqr N/A

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

   17. metadata-eval N/A

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

   18. pow-flip N/A

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

   19. lower-pow.f64 N/A

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

   20. metadata-eval 88.8

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

6. Applied rewrites 88.8%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
7. Add Preprocessing

Alternative 15: 6.3% accurate, 36.6× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 7.0%

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

   1. lift-+.f64 N/A

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

   2. add-flip N/A

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

   3. sub-to-mult N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. metadata-eval 7.0

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

3. Applied rewrites 7.0%

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

4. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   4. lower-cbrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   5. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, \sqrt[3]{x}, \frac{\frac{5}{81}}{{x}^{\frac{5}{3}}}\right) - \frac{\frac{1}{9}}{{x}^{\frac{2}{3}}}}{x} \]

   8. lower-pow.f64 98.4

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x} \]

6. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{x}, \frac{0.06172839506172839}{{x}^{1.6666666666666667}}\right) - \frac{0.1111111111111111}{{x}^{0.6666666666666666}}}{x}} \]

7. Taylor expanded in x around 0

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

9. Applied rewrites 6.3%

   \[\leadsto \color{blue}{1} \]
10. 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 2025142 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  :herbie-expected 5/2

  :alt
  (! :herbie-platform c (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

  (- (cbrt (+ x 1.0)) (cbrt x)))