2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.1%

Time: 9.2s

Alternatives: 6

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.5%

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

3. Taylor expanded in x around inf

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{9} \cdot \sqrt[3]{x} + \frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{-1}{9} \cdot \sqrt[3]{x}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\sqrt[3]{x} \cdot \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sqrt[3]{x}\right), \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \left(\frac{1}{3} \cdot \sqrt[3]{{x}^{4}}\right)\right), \left({x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{{x}^{4}}\right)\right)\right), \left({x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left({x}^{4}\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(x, 4\right)\right)\right)\right), \left({x}^{2}\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(x, 4\right)\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

   10. *-lowering-*.f64 21.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cbrt.f64}\left(x\right), \frac{-1}{9}\right), \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{pow.f64}\left(x, 4\right)\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

5. Simplified 21.3%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right), \color{blue}{\left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)}\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \left(\sqrt[3]{\frac{1}{{x}^{2}}}\right)\right), \left(\color{blue}{\frac{-1}{9}} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\left(\frac{1}{{x}^{2}}\right)\right)\right), \left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{2}\right)\right)\right)\right), \left(\frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\right) \]

   6. unpow2 N/A

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

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

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

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{3}, \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), \mathsf{*.f64}\left(\frac{-1}{9}, \color{blue}{\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right)}\right)\right) \]

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

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

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

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

   11. pow-lowering-pow.f64 47.9%

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

8. Simplified 47.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + -0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}} \]
9. Step-by-step derivation

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. pow1/3 N/A

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

   5. pow-prod-down N/A

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

   6. pow-prod-up N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{3} \cdot 1}{{x}^{\left(2 - \frac{4}{3}\right)}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   9. pow-div N/A

      \[\leadsto \frac{\frac{1}{3} \cdot 1}{\frac{{x}^{2}}{{x}^{\frac{4}{3}}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   10. pow2 N/A

       \[\leadsto \frac{\frac{1}{3} \cdot 1}{\frac{x \cdot x}{{x}^{\frac{4}{3}}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{3} \cdot 1}{\frac{x \cdot x}{{x}^{\left(4 \cdot \frac{1}{3}\right)}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   12. pow-pow N/A

       \[\leadsto \frac{\frac{1}{3} \cdot 1}{\frac{x \cdot x}{{\left({x}^{4}\right)}^{\frac{1}{3}}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   13. pow1/3 N/A

       \[\leadsto \frac{\frac{1}{3} \cdot 1}{\frac{x \cdot x}{\sqrt[3]{{x}^{4}}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   14. associate-/l* N/A

       \[\leadsto \frac{\frac{1}{3} \cdot 1}{x \cdot \frac{x}{\sqrt[3]{{x}^{4}}}} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   15. frac-times N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \frac{1}{\frac{x}{\sqrt[3]{{x}^{4}}}} + \color{blue}{\frac{-1}{9}} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   16. associate-/r/ N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \left(\frac{1}{x} \cdot \sqrt[3]{{x}^{4}}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   17. inv-pow N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \left({x}^{-1} \cdot \sqrt[3]{{x}^{4}}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   18. pow1/3 N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \left({x}^{-1} \cdot {\left({x}^{4}\right)}^{\frac{1}{3}}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   19. pow-pow N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \left({x}^{-1} \cdot {x}^{\left(4 \cdot \frac{1}{3}\right)}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot \left({x}^{-1} \cdot {x}^{\frac{4}{3}}\right) + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   21. pow-prod-up N/A

       \[\leadsto \frac{\frac{1}{3}}{x} \cdot {x}^{\left(-1 + \frac{4}{3}\right)} + \frac{-1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}} \]

   22. metadata-eval N/A

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

   23. pow1/3 N/A

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

10. Applied egg-rr 98.8%

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

Alternative 2: 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 5.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.3%

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

5. Simplified 47.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 89.6%

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

7. Applied egg-rr 89.6%

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. pow1/3 N/A

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

   4. inv-pow N/A

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

   5. div-inv N/A

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

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

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

   7. cbrt-lowering-cbrt.f64 98.4%

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

9. Applied egg-rr 98.4%

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

10. Final simplification 98.4%

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

Alternative 3: 97.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.3%

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

5. Simplified 47.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. un-div-inv N/A

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

   4. cbrt-prod N/A

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

   5. associate-/r* N/A

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

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

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

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

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

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

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

   9. cbrt-lowering-cbrt.f64 97.6%

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

7. Applied egg-rr 97.6%

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

   1. associate-/l/ N/A

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

   2. metadata-eval N/A

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

   3. pow1/3 N/A

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

   4. pow1/3 N/A

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

   5. pow-prod-up N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-div N/A

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

   9. pow2 N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. pow1/3 N/A

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

   13. associate-/l* N/A

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

   14. frac-times N/A

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

   15. clear-num N/A

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

   16. div-inv N/A

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

   17. pow1/3 N/A

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

   18. pow-pow N/A

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

   19. metadata-eval N/A

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

   20. inv-pow N/A

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

   21. pow-prod-up N/A

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

   22. metadata-eval N/A

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

   23. pow1/3 N/A

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

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

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

9. Applied egg-rr 98.3%

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

Alternative 4: 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 5.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.3%

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

5. Simplified 47.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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. un-div-inv N/A

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

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

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

   5. pow1/3 N/A

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

   6. pow2 N/A

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

   7. pow-pow N/A

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

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

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

   9. metadata-eval 89.6%

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

7. Applied egg-rr 89.6%

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

Alternative 5: 88.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 5.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.3%

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

5. Simplified 47.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 89.6%

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

7. Applied egg-rr 89.6%

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

8. Final simplification 89.6%

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

Alternative 6: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 2024164 
(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)))