2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 97.5%

Time: 3.3s

Alternatives: 12

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt[3]{x}}\\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, t\_0 \cdot \left(t\_0 \cdot 0.3333333333333333\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (cbrt x))))
   (fma
    (pow x -1.6666666666666667)
    -0.1111111111111111
    (* t_0 (* t_0 0.3333333333333333)))))

C

double code(double x) {
	double t_0 = 1.0 / cbrt(x);
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (t_0 * (t_0 * 0.3333333333333333)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / cbrt(x))
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64(t_0 * Float64(t_0 * 0.3333333333333333)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(t$95$0 * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. pow1/3 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   12. lower-*.f64 46.0

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 46.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-flip N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 89.8

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

7. Applied rewrites 89.8%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-cbrt.f64 97.5

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

9. Applied rewrites 97.5%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-cbrt.f64 N/A

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

    6. lift-pow.f64 N/A

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

    7. unpow-1 N/A

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

    8. lower-/.f64 N/A

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

    9. lift-cbrt.f64 N/A

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

    10. lower-*.f64 97.5

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

    11. lift-cbrt.f64 N/A

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

    12. lift-pow.f64 N/A

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

    13. unpow-1 N/A

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

    14. lower-/.f64 N/A

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

    15. lift-cbrt.f64 97.5

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

11. Applied rewrites 97.5%

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

Alternative 2: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (pow x -1.6666666666666667)
  -0.1111111111111111
  (* (pow (cbrt x) -2.0) 0.3333333333333333)))

C

double code(double x) {
	return fma(pow(x, -1.6666666666666667), -0.1111111111111111, (pow(cbrt(x), -2.0) * 0.3333333333333333));
}

Julia

function code(x)
	return fma((x ^ -1.6666666666666667), -0.1111111111111111, Float64((cbrt(x) ^ -2.0) * 0.3333333333333333))
end

Wolfram

code[x_] := N[(N[Power[x, -1.6666666666666667], $MachinePrecision] * -0.1111111111111111 + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{{x}^{2}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}}{\color{blue}{{x}^{2}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \sqrt[3]{{x}^{4}} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{\color{blue}{x}}^{2}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{3} + \frac{-1}{9} \cdot \sqrt[3]{x}}{{x}^{2}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{\color{blue}{x}}^{2}} \]

   5. pow1/3 N/A

      \[\leadsto \frac{\mathsf{fma}\left({\left({x}^{4}\right)}^{\frac{1}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   6. pow-pow N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\left(4 \cdot \frac{1}{3}\right)}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   10. lift-cbrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left({x}^{\frac{4}{3}}, \frac{1}{3}, \frac{-1}{9} \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

   12. lower-*.f64 46.0

       \[\leadsto \frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

4. Applied rewrites 46.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({x}^{1.3333333333333333}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. pow1/3 N/A

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

   14. pow-flip N/A

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

   15. pow-pow N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-*.f64 89.8

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

7. Applied rewrites 89.8%

   \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, \color{blue}{-0.1111111111111111}, {x}^{-0.6666666666666666} \cdot 0.3333333333333333\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. metadata-eval N/A

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

   5. pow-sqr N/A

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lift-cbrt.f64 97.5

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

9. Applied rewrites 97.5%

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

    1. lift-*.f64 N/A

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

    2. lift-cbrt.f64 N/A

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

    3. lift-pow.f64 N/A

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

    4. lift-cbrt.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. pow-prod-up N/A

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

    7. lower-pow.f64 N/A

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

    8. lift-cbrt.f64 N/A

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

    9. metadata-eval 97.5

       \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]

11. Applied rewrites 97.5%

    \[\leadsto \mathsf{fma}\left({x}^{-1.6666666666666667}, -0.1111111111111111, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 3: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.9

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

4. Applied rewrites 88.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

   9. cbrt-div N/A

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

   10. metadata-eval N/A

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

   11. rem-cbrt-cube N/A

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

   12. lower-/.f64 N/A

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

   13. lower-cbrt.f64 N/A

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

   14. pow2 N/A

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

   15. distribute-lft-neg-in N/A

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

   16. lower-*.f64 N/A

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

   17. lower-neg.f64 50.4

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

6. Applied rewrites 50.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-neg.f64 N/A

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

   8. cbrt-prod N/A

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

   9. lift-neg.f64 N/A

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. cbrt-unprod N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. cbrt-unprod N/A

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

   17. *-commutative N/A

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

   18. mul-1-neg N/A

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

   19. lift-neg.f64 N/A

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

   20. lower-cbrt.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. lift-cbrt.f64 96.5

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

8. Applied rewrites 96.5%

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

Alternative 4: 96.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.9

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

4. Applied rewrites 88.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. pow2 N/A

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

   10. cbrt-prod N/A

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

   11. lower-/.f64 N/A

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

   12. pow1/3 N/A

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

   13. pow1/3 N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

   18. metadata-eval 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-cbrt.f64 N/A

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

   8. lift-cbrt.f64 96.5

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

8. Applied rewrites 96.5%

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

Alternative 5: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(\sqrt[3]{x}\right)}^{2}} \cdot 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 1.0 (pow (cbrt x) 2.0)) 0.3333333333333333))

C

double code(double x) {
	return (1.0 / pow(cbrt(x), 2.0)) * 0.3333333333333333;
}

Java

public static double code(double x) {
	return (1.0 / Math.pow(Math.cbrt(x), 2.0)) * 0.3333333333333333;
}

Julia

function code(x)
	return Float64(Float64(1.0 / (cbrt(x) ^ 2.0)) * 0.3333333333333333)
end

Wolfram

code[x_] := N[(N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.9

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

4. Applied rewrites 88.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. pow-flip N/A

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

   6. pow1/3 N/A

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

   7. cbrt-div N/A

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

   8. metadata-eval N/A

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

   9. pow2 N/A

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

   10. cbrt-prod N/A

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

   11. lower-/.f64 N/A

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

   12. pow1/3 N/A

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

   13. pow1/3 N/A

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

   14. pow-prod-up N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

       \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

   17. lower-pow.f64 N/A

       \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

   18. metadata-eval 88.9

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

6. Applied rewrites 88.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. pow1/3 N/A

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

   5. pow1/3 N/A

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

   6. pow2 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-cbrt.f64 96.5

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

8. Applied rewrites 96.5%

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

Alternative 6: 92.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x \cdot x}} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\frac{\log x \cdot 2}{3}}} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* (/ 1.0 (cbrt (* x x))) 0.3333333333333333)
   (* (/ 1.0 (exp (/ (* (log x) 2.0) 3.0))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / exp(((log(x) * 2.0) / 3.0))) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / Math.exp(((Math.log(x) * 2.0) / 3.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / exp(Float64(Float64(log(x) * 2.0) / 3.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[Exp[N[(N[(N[Log[x], $MachinePrecision] * 2.0), $MachinePrecision] / 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      18. metadata-eval 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      18. metadata-eval 89.1

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

   6. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. pow-to-exp N/A

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

      6. exp-cbrt-rev N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 90.0

          \[\leadsto \frac{1}{e^{\frac{\log x \cdot 2}{3}}} \cdot 0.3333333333333333 \]

   8. Applied rewrites 90.0%

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

Alternative 7: 92.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      18. metadata-eval 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      18. metadata-eval 89.1

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

   6. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-log.f64 89.6

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

   8. Applied rewrites 89.6%

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

Alternative 8: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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 < 1.35000000000000003e154

   1. Initial program 9.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. cbrt-div N/A

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

      8. metadata-eval N/A

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

      9. pow2 N/A

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

      10. cbrt-prod N/A

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

      11. lower-/.f64 N/A

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

      12. pow1/3 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      17. lower-pow.f64 N/A

          \[\leadsto \frac{1}{{x}^{\left(\frac{\frac{4}{3}}{2}\right)}} \cdot \frac{1}{3} \]

      18. metadata-eval 88.6

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

   6. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow1/3 N/A

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

      5. lower-cbrt.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-log.f64 89.6

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

   6. Applied rewrites 89.6%

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

Alternative 9: 92.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+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 <= 1.35e+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 <= 1.35e+154)
		tmp = Float64(-0.3333333333333333 / cbrt(Float64(Float64(-x) * x)));
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+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 < 1.35000000000000003e154

   1. Initial program 9.0%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. pow-flip N/A

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

      6. pow1/3 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. cbrt-div N/A

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

      10. metadata-eval N/A

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

      11. rem-cbrt-cube N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cbrt.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-lft-neg-in N/A

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 95.2

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

   6. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. associate-*l/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-cbrt.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow1/3 N/A

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

      4. pow-flip N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval 89.1

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

   4. Applied rewrites 89.1%

      \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-log.f64 89.6

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

   6. Applied rewrites 89.6%

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

Alternative 10: 89.2% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333))

C

double code(double x) {
	return exp((log(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 = exp((log(x) * (-0.6666666666666666d0))) * 0.3333333333333333d0
end function

Java

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

Python

def code(x):
	return math.exp((math.log(x) * -0.6666666666666666)) * 0.3333333333333333

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.9

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

4. Applied rewrites 88.9%

   \[\leadsto \color{blue}{{x}^{-0.6666666666666666} \cdot 0.3333333333333333} \]
5. 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. lower-*.f64 N/A

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

   5. lower-log.f64 89.2

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

6. Applied rewrites 89.2%

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

Alternative 11: 88.9% 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 6.9%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval 88.9

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

4. Applied rewrites 88.9%

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

Alternative 12: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
5. 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 2025115 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :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)))