2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 97.1%

Time: 9.4s

Alternatives: 4

Speedup: 2.0×

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: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 48.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 48.1%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{x \cdot x}}\right), \frac{1}{3}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt[3]{x \cdot x}}\right), \frac{1}{3}\right) \]

   5. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{\left(x \cdot x\right)}^{\frac{1}{3}}}\right), \frac{1}{3}\right) \]

   6. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}\right), \frac{1}{3}\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{1}{3}\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

   11. metadata-eval 89.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

7. Applied egg-rr 89.8%

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \frac{2}{3}\right)}\right), \frac{1}{3}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \left(2 \cdot \frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   3. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{-1}\right)}^{\left(2 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x}\right)}^{\left(2 \cdot \frac{1}{3}\right)}\right), \frac{1}{3}\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\frac{1}{x}\right)}^{2}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x} \cdot \frac{1}{x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\frac{1}{x}}{x}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   8. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}\right)}^{\frac{1}{3}}\right), \frac{1}{3}\right) \]

   9. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right), \frac{1}{3}\right) \]

   10. cbrt-undiv N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\sqrt[3]{\mathsf{neg}\left(x\right)}}\right), \frac{1}{3}\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{\sqrt[3]{\mathsf{neg}\left(x\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}}\right), \frac{1}{3}\right) \]

   12. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\sqrt[3]{\mathsf{neg}\left(x\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}\right)}^{-1}\right), \frac{1}{3}\right) \]

   13. pow-to-exp N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left(\frac{\sqrt[3]{\mathsf{neg}\left(x\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}\right) \cdot -1}\right), \frac{1}{3}\right) \]

   14. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\frac{\sqrt[3]{\mathsf{neg}\left(x\right)}}{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   15. cbrt-undiv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\sqrt[3]{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\frac{1}{x}\right)}}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   16. frac-2neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\sqrt[3]{\frac{x}{\frac{1}{x}}}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   17. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\sqrt[3]{x \cdot \frac{1}{\frac{1}{x}}}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   18. remove-double-div N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\sqrt[3]{x \cdot x}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   19. cbrt-unprod N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot -1\right)\right), \frac{1}{3}\right) \]

   20. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right), -1\right)\right), \frac{1}{3}\right) \]

9. Applied egg-rr 90.4%

   \[\leadsto \color{blue}{e^{\log \left({x}^{0.6666666666666666}\right) \cdot -1}} \cdot 0.3333333333333333 \]
10. Step-by-step derivation

    1. exp-to-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{\frac{2}{3}}\right)}^{-1}\right), \frac{1}{3}\right) \]

    2. inv-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{x}^{\frac{2}{3}}}\right), \frac{1}{3}\right) \]

    3. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{x}^{\left(2 - \frac{4}{3}\right)}}\right), \frac{1}{3}\right) \]

    4. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{x}^{\left(2 - 4 \cdot \frac{1}{3}\right)}}\right), \frac{1}{3}\right) \]

    5. pow-div N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{{x}^{2}}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}}\right), \frac{1}{3}\right) \]

    6. pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}}\right), \frac{1}{3}\right) \]

    7. pow-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x}{{\left({x}^{4}\right)}^{\frac{1}{3}}}}\right), \frac{1}{3}\right) \]

    8. pow1/3 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{x \cdot x}{\sqrt[3]{{x}^{4}}}}\right), \frac{1}{3}\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt[3]{{x}^{4}}}{x \cdot x}\right), \frac{1}{3}\right) \]

    10. associate-/r* N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\sqrt[3]{{x}^{4}}}{x}}{x}\right), \frac{1}{3}\right) \]

    11. div-inv N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt[3]{{x}^{4}} \cdot \frac{1}{x}}{x}\right), \frac{1}{3}\right) \]

    12. pow1/3 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{\left({x}^{4}\right)}^{\frac{1}{3}} \cdot \frac{1}{x}}{x}\right), \frac{1}{3}\right) \]

    13. pow-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\left(4 \cdot \frac{1}{3}\right)} \cdot \frac{1}{x}}{x}\right), \frac{1}{3}\right) \]

    14. inv-pow N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\left(4 \cdot \frac{1}{3}\right)} \cdot {x}^{-1}}{x}\right), \frac{1}{3}\right) \]

    15. pow-prod-up N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\left(4 \cdot \frac{1}{3} + -1\right)}}{x}\right), \frac{1}{3}\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\left(\frac{4}{3} + -1\right)}}{x}\right), \frac{1}{3}\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\frac{1}{3}}}{x}\right), \frac{1}{3}\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{{x}^{\left(\frac{\frac{2}{3}}{2}\right)}}{x}\right), \frac{1}{3}\right) \]

    19. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{\frac{2}{3}}{2}\right)}\right), x\right), \frac{1}{3}\right) \]

    20. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x}^{\frac{1}{3}}\right), x\right), \frac{1}{3}\right) \]

    21. unpow1/3 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt[3]{x}\right), x\right), \frac{1}{3}\right) \]

    22. cbrt-lowering-cbrt.f64 98.5%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(x\right), x\right), \frac{1}{3}\right) \]

11. Applied egg-rr 98.5%

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

Alternative 2: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 48.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 48.1%

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

   1. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\frac{1}{x}}{x}}\right)\right) \]

   2. frac-2neg N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{\mathsf{neg}\left(\frac{1}{x}\right)}{\mathsf{neg}\left(x\right)}}\right)\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\frac{\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}}{\color{blue}{\sqrt[3]{\mathsf{neg}\left(x\right)}}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\left(\sqrt[3]{\mathsf{neg}\left(\frac{1}{x}\right)}\right), \color{blue}{\left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)}\right)\right) \]

   5. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\color{blue}{\mathsf{neg}\left(x\right)}}\right)\right)\right) \]

   6. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left(\frac{1}{x}\right)\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(\color{blue}{x}\right)}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \left(\sqrt[3]{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   8. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{cbrt.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right) \]

   9. neg-lowering-neg.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

7. Applied egg-rr 97.8%

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

   1. inv-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left({x}^{-1}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   2. rem-cube-cbrt N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left({\left({\left(\sqrt[3]{x}\right)}^{3}\right)}^{-1}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   4. unpow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1} \cdot {\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   6. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   7. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({\left({x}^{\frac{1}{3}}\right)}^{\left(\frac{3}{2}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\left(\frac{1}{3} \cdot \frac{3}{2}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\left(\frac{1}{3} \cdot \frac{3}{2}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\frac{1}{2}}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\left(3 \cdot \frac{1}{6}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({x}^{\left(3 \cdot \frac{\frac{1}{3}}{2}\right)}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   13. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \left(3 \cdot \frac{\frac{1}{3}}{2}\right)\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \left(3 \cdot \frac{1}{6}\right)\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{2}\right), -1\right), \left({\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right)}^{-1}\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

   16. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{pow.f64}\left(x, \frac{1}{2}\right), -1\right), \mathsf{pow.f64}\left(\left({\left(\sqrt[3]{x}\right)}^{\left(\frac{3}{2}\right)}\right), -1\right)\right)\right)\right), \mathsf{cbrt.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right)\right) \]

9. Applied egg-rr 97.8%

   \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt[3]{-\color{blue}{{\left({x}^{0.5}\right)}^{-1} \cdot {\left({x}^{0.5}\right)}^{-1}}}}{\sqrt[3]{-x}} \]
10. Step-by-step derivation

    1. cbrt-undiv N/A

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

    2. pow1/3 N/A

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

    3. frac-2neg N/A

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

    4. div-inv N/A

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

    5. pow-prod-up N/A

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

    6. pow-pow N/A

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

    7. metadata-eval N/A

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

    8. metadata-eval N/A

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

    9. inv-pow N/A

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

    10. unpow-prod-down N/A

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

    11. unpow-prod-down N/A

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

    12. pow-prod-up N/A

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

    13. pow-pow N/A

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

    14. metadata-eval N/A

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

    15. metadata-eval N/A

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

    16. metadata-eval N/A

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

    17. metadata-eval N/A

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

    18. pow-div N/A

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

    19. pow-pow N/A

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

    20. pow1/3 N/A

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

    21. pow2 N/A

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

    22. div-inv N/A

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

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

Alternative 3: 88.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{-1 \cdot -1}{{x}^{2}}}\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{-1 \cdot \frac{-1}{{x}^{2}}}\right)\right) \]

   4. cbrt-lowering-cbrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(-1 \cdot \frac{-1}{{x}^{2}}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{-1 \cdot -1}{{x}^{2}}\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right)\right) \]

   9. *-lowering-*.f64 48.1%

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right) \]

5. Simplified 48.1%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}}\right), \color{blue}{\frac{1}{3}}\right) \]

   3. cbrt-div N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt[3]{1}}{\sqrt[3]{x \cdot x}}\right), \frac{1}{3}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt[3]{x \cdot x}}\right), \frac{1}{3}\right) \]

   5. pow1/3 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{{\left(x \cdot x\right)}^{\frac{1}{3}}}\right), \frac{1}{3}\right) \]

   6. pow-flip N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   7. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right), \frac{1}{3}\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}\right), \frac{1}{3}\right) \]

   9. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{1}{3}\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(2 \cdot \frac{-1}{3}\right)\right), \frac{1}{3}\right) \]

   11. metadata-eval 89.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-2}{3}\right), \frac{1}{3}\right) \]

7. Applied egg-rr 89.8%

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

8. Final simplification 89.8%

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

Alternative 4: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\sqrt[3]{x}\right)}\right) \]

   2. cbrt-lowering-cbrt.f64 1.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{cbrt.f64}\left(x\right)\right) \]

5. Simplified 1.8%

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

Developer Target 1: 98.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024140 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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