2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 98.5%

Time: 13.2s

Alternatives: 7

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: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\left(0.012345679012345678 + \frac{0.0038103947568968147}{x \cdot x}\right) \cdot \sqrt[3]{x}}{-0.1111111111111111 + \frac{-0.06172839506172839}{x}}}{x} - -0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{\frac{-1}{x}}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (/
    (/
     (* (+ 0.012345679012345678 (/ 0.0038103947568968147 (* x x))) (cbrt x))
     (+ -0.1111111111111111 (/ -0.06172839506172839 x)))
    x)
   (* -0.3333333333333333 (cbrt (/ -1.0 (/ -1.0 x)))))
  x))

C

double code(double x) {
	return (((((0.012345679012345678 + (0.0038103947568968147 / (x * x))) * cbrt(x)) / (-0.1111111111111111 + (-0.06172839506172839 / x))) / x) - (-0.3333333333333333 * cbrt((-1.0 / (-1.0 / x))))) / x;
}

Java

public static double code(double x) {
	return (((((0.012345679012345678 + (0.0038103947568968147 / (x * x))) * Math.cbrt(x)) / (-0.1111111111111111 + (-0.06172839506172839 / x))) / x) - (-0.3333333333333333 * Math.cbrt((-1.0 / (-1.0 / x))))) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.012345679012345678 + Float64(0.0038103947568968147 / Float64(x * x))) * cbrt(x)) / Float64(-0.1111111111111111 + Float64(-0.06172839506172839 / x))) / x) - Float64(-0.3333333333333333 * cbrt(Float64(-1.0 / Float64(-1.0 / x))))) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.012345679012345678 + N[(0.0038103947568968147 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[(-0.1111111111111111 + N[(-0.06172839506172839 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-0.3333333333333333 * N[Power[N[(-1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 7.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

4. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-55}{648} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{5}{216} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \left(\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\right)}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x}} \]

7. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{\frac{-1}{x}}} - \frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.1111111111111111 - \frac{-0.06172839506172839}{x}\right)}{x}}{0 - x}} \]

9. Applied egg-rr 97.9%

   \[\leadsto \frac{-0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{\frac{-1}{x}}} - \frac{\color{blue}{\frac{\left(0.012345679012345678 + \frac{0.0038103947568968147}{x \cdot x}\right) \cdot \sqrt[3]{x}}{-0.1111111111111111 + \frac{-0.06172839506172839}{x}}}}{x}}{0 - x} \]

10. Final simplification 97.9%

    \[\leadsto \frac{\frac{\frac{\left(0.012345679012345678 + \frac{0.0038103947568968147}{x \cdot x}\right) \cdot \sqrt[3]{x}}{-0.1111111111111111 + \frac{-0.06172839506172839}{x}}}{x} - -0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{\frac{-1}{x}}}}{x} \]
11. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.1111111111111111 - \frac{-0.06172839506172839}{x}\right)}{x} - -0.3333333333333333 \cdot \sqrt[3]{x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (-
   (/
    (*
     (cbrt (/ -1.0 (/ -1.0 x)))
     (- -0.1111111111111111 (/ -0.06172839506172839 x)))
    x)
   (* -0.3333333333333333 (cbrt x)))
  x))

C

double code(double x) {
	return (((cbrt((-1.0 / (-1.0 / x))) * (-0.1111111111111111 - (-0.06172839506172839 / x))) / x) - (-0.3333333333333333 * cbrt(x))) / x;
}

Java

public static double code(double x) {
	return (((Math.cbrt((-1.0 / (-1.0 / x))) * (-0.1111111111111111 - (-0.06172839506172839 / x))) / x) - (-0.3333333333333333 * Math.cbrt(x))) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(cbrt(Float64(-1.0 / Float64(-1.0 / x))) * Float64(-0.1111111111111111 - Float64(-0.06172839506172839 / x))) / x) - Float64(-0.3333333333333333 * cbrt(x))) / x)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[Power[N[(-1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(-0.1111111111111111 - N[(-0.06172839506172839 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(-0.3333333333333333 * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 7.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

4. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-55}{648} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{5}{216} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \left(\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\right)}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x}} \]

7. Simplified 97.9%

   \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \sqrt[3]{\frac{-1}{\frac{-1}{x}}} - \frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.1111111111111111 - \frac{-0.06172839506172839}{x}\right)}{x}}{0 - x}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt[3]{\frac{-1}{-1} \cdot x} \cdot \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt[3]{1 \cdot x} \cdot \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

   4. *-lft-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt[3]{x} \cdot \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{x}\right), \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

   6. cbrt-lowering-cbrt.f64 97.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{3}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(-1, x\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{9}, \mathsf{/.f64}\left(\frac{-5}{81}, x\right)\right)\right), x\right)\right), \mathsf{\_.f64}\left(0, x\right)\right) \]

9. Applied egg-rr 97.9%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{x} \cdot -0.3333333333333333} - \frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.1111111111111111 - \frac{-0.06172839506172839}{x}\right)}{x}}{0 - x} \]

10. Final simplification 97.9%

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.1111111111111111 - \frac{-0.06172839506172839}{x}\right)}{x} - -0.3333333333333333 \cdot \sqrt[3]{x}}{x} \]
11. Add Preprocessing

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 7.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

4. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)}{x}} \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(\frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{3} \cdot \sqrt[3]{x}\right)\right), \color{blue}{x}\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) + \frac{1}{3} \cdot \sqrt[3]{x}\right), x\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{3} \cdot \sqrt[3]{x} + \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \sqrt[3]{x}\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x}\right)\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\frac{-5}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \frac{1}{36} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), x\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \left(\frac{-5}{36} + \frac{1}{36}\right)\right)\right), x\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \left(\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \frac{-1}{9}\right)\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\left(\sqrt[3]{\frac{1}{{x}^{2}}}\right), \frac{-1}{9}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right), \frac{-1}{9}\right)\right), x\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right), \frac{-1}{9}\right)\right), x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left(x \cdot x\right)\right)\right), \frac{-1}{9}\right)\right), x\right) \]

   13. *-lowering-*.f64 97.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right), \mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right), \frac{-1}{9}\right)\right), x\right) \]

7. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{x \cdot x}} \cdot -0.1111111111111111}{x}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{-1}{9} + \frac{1}{3} \cdot \sqrt[3]{x}\right), x\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\sqrt[3]{\frac{1}{x \cdot x}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   3. pow1/3 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(\frac{1}{x \cdot x}\right)}^{\frac{1}{3}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   4. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left({\left(x \cdot x\right)}^{-1}\right)}^{\frac{1}{3}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   5. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left(x \cdot x\right)}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left({x}^{2}\right)}^{\left(-1 \cdot \frac{1}{3}\right)} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({\left({x}^{2}\right)}^{\frac{-1}{3}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   8. pow-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({x}^{\left(2 \cdot \frac{-1}{3}\right)} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({x}^{\frac{-2}{3}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left({x}^{\left(\mathsf{neg}\left(\frac{2}{3}\right)\right)} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   11. pow-flip N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{{x}^{\frac{2}{3}}} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   12. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1 \cdot \frac{-1}{9}}{{x}^{\frac{2}{3}}}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{9}}{{x}^{\frac{2}{3}}}\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{9}, \left({x}^{\frac{2}{3}}\right)\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{9}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right), \left(\frac{1}{3} \cdot \sqrt[3]{x}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{9}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{x}\right)\right)\right), x\right) \]

   17. cbrt-lowering-cbrt.f64 97.4%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{9}, \mathsf{pow.f64}\left(x, \frac{2}{3}\right)\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(x\right)\right)\right), x\right) \]

9. Applied egg-rr 97.4%

   \[\leadsto \frac{\color{blue}{\frac{-0.1111111111111111}{{x}^{0.6666666666666666}} + 0.3333333333333333 \cdot \sqrt[3]{x}}}{x} \]

10. Final simplification 97.4%

    \[\leadsto \frac{\frac{-0.1111111111111111}{{x}^{0.6666666666666666}} + \sqrt[3]{x} \cdot 0.3333333333333333}{x} \]
11. Add Preprocessing

Alternative 4: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(\frac{-0.1111111111111111}{x} - -0.3333333333333333\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (cbrt (/ -1.0 (/ -1.0 x)))
   (- (/ -0.1111111111111111 x) -0.3333333333333333))
  x))

C

double code(double x) {
	return (cbrt((-1.0 / (-1.0 / x))) * ((-0.1111111111111111 / x) - -0.3333333333333333)) / x;
}

Java

public static double code(double x) {
	return (Math.cbrt((-1.0 / (-1.0 / x))) * ((-0.1111111111111111 / x) - -0.3333333333333333)) / x;
}

Julia

function code(x)
	return Float64(Float64(cbrt(Float64(-1.0 / Float64(-1.0 / x))) * Float64(Float64(-0.1111111111111111 / x) - -0.3333333333333333)) / x)
end

Wolfram

code[x_] := N[(N[(N[Power[N[(-1.0 / N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[(-0.1111111111111111 / x), $MachinePrecision] - -0.3333333333333333), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. pow1/3 N/A

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

   2. sqr-pow N/A

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

   3. pow1/3 N/A

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

   4. sqr-pow N/A

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

   5. difference-of-squares N/A

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

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

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

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

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

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

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

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

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

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right), \left({\left(x + \color{blue}{1}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \left(\frac{\frac{1}{3}}{2}\right)\right)\right), \left({\left(x + 1\right)}^{\color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{\color{blue}{2}}\right)} - {x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\left({\left(x + 1\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right), \color{blue}{\left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\left(x + 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({\color{blue}{x}}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(\frac{\frac{1}{3}}{2}\right)\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \left({x}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{3}}{2}\right)}\right)\right)\right) \]

   18. metadata-eval 7.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right), \mathsf{\_.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{6}\right), \mathsf{pow.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

4. Applied egg-rr 7.9%

   \[\leadsto \color{blue}{\left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - {x}^{0.16666666666666666}\right)} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x}\right) \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{\color{blue}{\mathsf{neg}\left(x\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{-1 \cdot \frac{\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{-1 \cdot \color{blue}{x}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot \frac{\frac{-5}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2} + \frac{1}{36} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}}{x} + \frac{-1}{3} \cdot {\left(e^{\frac{1}{6} \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{x}\right)\right)}\right)}^{2}\right), \color{blue}{\left(-1 \cdot x\right)}\right) \]

7. Simplified 97.4%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(-0.3333333333333333 - \frac{-0.1111111111111111}{x}\right)}{0 - x}} \]

8. Final simplification 97.4%

   \[\leadsto \frac{\sqrt[3]{\frac{-1}{\frac{-1}{x}}} \cdot \left(\frac{-0.1111111111111111}{x} - -0.3333333333333333\right)}{x} \]
9. Add Preprocessing

Alternative 5: 97.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around inf

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

   1. *-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 44.9%

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

5. Simplified 44.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 87.9%

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

7. Applied egg-rr 87.9%

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

   1. rem-cube-cbrt N/A

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

   2. pow-pow N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

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

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

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

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

   7. metadata-eval 95.7%

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

9. Applied egg-rr 95.7%

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

    1. pow1/3 N/A

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

    2. pow-pow N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

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

    5. pow-div N/A

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

    6. pow1/3 N/A

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

    7. unpow1 N/A

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

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

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

    9. cbrt-lowering-cbrt.f64 96.4%

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

11. Applied egg-rr 96.4%

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

12. Final simplification 96.4%

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

Alternative 6: 88.9% 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 7.9%

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

3. Taylor expanded in x around inf

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

   1. *-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 44.9%

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

5. Simplified 44.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

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

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

   9. metadata-eval N/A

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

   10. metadata-eval 87.9%

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

7. Applied egg-rr 87.9%

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

8. Final simplification 87.9%

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

Alternative 7: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

3. Taylor expanded in x around 0

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

   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: 98.5% 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 2024191 
(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)))