2cbrt (problem 3.3.4)

Percentage Accurate: 7.0% → 98.0%

Time: 10.3s

Alternatives: 8

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: 7.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{x}{\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + \sqrt[3]{x} \cdot -0.3333333333333333}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  -1.0
  (/
   x
   (+
    (/ 0.1111111111111111 (pow x 0.6666666666666666))
    (* (cbrt x) -0.3333333333333333)))))

C

double code(double x) {
	return -1.0 / (x / ((0.1111111111111111 / pow(x, 0.6666666666666666)) + (cbrt(x) * -0.3333333333333333)));
}

Java

public static double code(double x) {
	return -1.0 / (x / ((0.1111111111111111 / Math.pow(x, 0.6666666666666666)) + (Math.cbrt(x) * -0.3333333333333333)));
}

Julia

function code(x)
	return Float64(-1.0 / Float64(x / Float64(Float64(0.1111111111111111 / (x ^ 0.6666666666666666)) + Float64(cbrt(x) * -0.3333333333333333))))
end

Wolfram

code[x_] := N[(-1.0 / N[(x / N[(N[(0.1111111111111111 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 6.5%

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

4. Applied egg-rr 6.5%

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

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

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

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{x \cdot x}} \cdot -0.1111111111111111}{x}} \]

9. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + \sqrt[3]{x} \cdot -0.3333333333333333}}} \]
10. Add Preprocessing

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 6.5%

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

4. Applied egg-rr 6.5%

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

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

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

7. Simplified 98.2%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

   3. cbrt-div N/A

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

   4. cbrt-div N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. un-div-inv N/A

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

   8. associate-/r* N/A

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

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

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

   10. pow1/3 N/A

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

   11. pow1/3 N/A

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

   12. pow-sqr N/A

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

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

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

   14. metadata-eval 98.2%

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

9. Applied egg-rr 98.2%

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

10. Final simplification 98.2%

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

Alternative 3: 92.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.0%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 95.8%

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

   5. Simplified 95.8%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

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

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

      10. metadata-eval 89.1%

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

   7. Applied egg-rr 89.1%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. pow1/3 N/A

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

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

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

      6. *-lowering-*.f64 96.0%

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

   9. Applied egg-rr 96.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.7%

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

   5. Simplified 4.7%

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

      1. cbrt-div N/A

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

      2. metadata-eval N/A

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

      3. cbrt-prod N/A

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

      4. un-div-inv N/A

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

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

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

      6. pow1/3 N/A

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

      7. pow1/3 N/A

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

      8. pow-prod-up N/A

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

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

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

      10. metadata-eval 89.2%

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

   7. Applied egg-rr 89.2%

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

Alternative 4: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 6.5%

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

4. Applied egg-rr 6.5%

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

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

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

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{x \cdot x}} \cdot -0.1111111111111111}{x}} \]

9. Applied egg-rr 98.3%

   \[\leadsto \color{blue}{\frac{-1}{\frac{x}{\frac{0.1111111111111111}{{x}^{0.6666666666666666}} + \sqrt[3]{x} \cdot -0.3333333333333333}}} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-1}{3} \cdot \sqrt[3]{x}\right)}\right)\right) \]
11. Step-by-step derivation

    1. *-commutative N/A

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

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

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

    3. cbrt-lowering-cbrt.f64 97.4%

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

12. Simplified 97.4%

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

Alternative 5: 97.1% accurate, 2.0× speedup

Math

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

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(Float64(cbrt(x) * 0.3333333333333333) / x)
end

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 6.5%

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

4. Applied egg-rr 6.5%

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

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

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

7. Simplified 98.2%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \sqrt[3]{x} + \sqrt[3]{\frac{1}{x \cdot x}} \cdot -0.1111111111111111}{x}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{3} \cdot \sqrt[3]{x}\right)}, x\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

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

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

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

   3. cbrt-lowering-cbrt.f64 97.4%

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

10. Simplified 97.4%

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

Alternative 6: 88.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.2%

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

5. Simplified 54.2%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

   4. un-div-inv N/A

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

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

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

   6. pow1/3 N/A

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

   7. pow1/3 N/A

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

   8. pow-prod-up N/A

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

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

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

   10. metadata-eval 89.1%

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

7. Applied egg-rr 89.1%

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

Alternative 7: 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 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 54.2%

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

5. Simplified 54.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 89.1%

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

7. Applied egg-rr 89.1%

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

8. Final simplification 89.1%

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

Alternative 8: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024150 
(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)))