2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.2%

Time: 11.3s

Alternatives: 11

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: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x} \cdot 0.6666666666666666}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \frac{1}{\sqrt[3]{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (/ 1.0 x) 0.6666666666666666)
  (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x)))) (/ 1.0 (cbrt x)))))

C

double code(double x) {
	return ((1.0 / x) * 0.6666666666666666) / (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / cbrt(x)));
}

Java

public static double code(double x) {
	return ((1.0 / x) * 0.6666666666666666) / (Math.cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / Math.cbrt(x)));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / x) * 0.6666666666666666) / Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(1.0 / cbrt(x))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] / N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

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

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

4. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left({\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. /-lowering-/.f64 97.7%

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

7. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}}} \]
8. Step-by-step derivation

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

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

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

   4. cbrt-lowering-cbrt.f64 97.9%

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

9. Applied egg-rr 97.9%

   \[\leadsto \frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \color{blue}{\frac{1}{\sqrt[3]{x}}}} \]
10. Step-by-step derivation

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

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

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

    4. /-lowering-/.f64 97.9%

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

11. Applied egg-rr 97.9%

    \[\leadsto \frac{\color{blue}{\frac{1}{x} \cdot 0.6666666666666666}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \frac{1}{\sqrt[3]{x}}} \]
12. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \frac{1}{\sqrt[3]{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ 0.6666666666666666 x)
  (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x)))) (/ 1.0 (cbrt x)))))

C

double code(double x) {
	return (0.6666666666666666 / x) / (cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / cbrt(x)));
}

Java

public static double code(double x) {
	return (0.6666666666666666 / x) / (Math.cbrt(((1.0 / x) + (2.0 / (x * x)))) + (1.0 / Math.cbrt(x)));
}

Julia

function code(x)
	return Float64(Float64(0.6666666666666666 / x) / Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + Float64(1.0 / cbrt(x))))
end

Wolfram

code[x_] := N[(N[(0.6666666666666666 / x), $MachinePrecision] / N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

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

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

4. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left({\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. /-lowering-/.f64 97.7%

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

7. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}}} \]
8. Step-by-step derivation

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

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

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

   4. cbrt-lowering-cbrt.f64 97.9%

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

9. Applied egg-rr 97.9%

   \[\leadsto \frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \color{blue}{\frac{1}{\sqrt[3]{x}}}} \]
10. Add Preprocessing

Alternative 3: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (/ 0.6666666666666666 x)
  (+ (cbrt (+ (/ 1.0 x) (/ 2.0 (* x x)))) (cbrt (/ 1.0 x)))))

C

double code(double x) {
	return (0.6666666666666666 / x) / (cbrt(((1.0 / x) + (2.0 / (x * x)))) + cbrt((1.0 / x)));
}

Java

public static double code(double x) {
	return (0.6666666666666666 / x) / (Math.cbrt(((1.0 / x) + (2.0 / (x * x)))) + Math.cbrt((1.0 / x)));
}

Julia

function code(x)
	return Float64(Float64(0.6666666666666666 / x) / Float64(cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / Float64(x * x)))) + cbrt(Float64(1.0 / x))))
end

Wolfram

code[x_] := N[(N[(0.6666666666666666 / x), $MachinePrecision] / N[(N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

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

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

4. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left({\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. /-lowering-/.f64 97.7%

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

7. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}}} \]
8. Add Preprocessing

Alternative 4: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\frac{1}{x}}\\ \frac{\frac{0.6666666666666666}{x}}{t\_0 + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (/ 1.0 x)))) (/ (/ 0.6666666666666666 x) (+ t_0 t_0))))

C

double code(double x) {
	double t_0 = cbrt((1.0 / x));
	return (0.6666666666666666 / x) / (t_0 + t_0);
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 / x));
	return (0.6666666666666666 / x) / (t_0 + t_0);
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 / x))
	return Float64(Float64(0.6666666666666666 / x) / Float64(t_0 + t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(0.6666666666666666 / x), $MachinePrecision] / N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.2%

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

   1. flip-- N/A

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

   2. flip-- N/A

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

   3. associate-/l/ N/A

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

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

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

4. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{\frac{{\left(x + 1\right)}^{1.3333333333333333} - {x}^{1.3333333333333333}}{\left({\left(x + 1\right)}^{0.3333333333333333} + \sqrt[3]{x}\right) \cdot \left({\left(x + 1\right)}^{0.6666666666666666} + {x}^{0.6666666666666666}\right)}} \]

5. Taylor expanded in x around inf

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

   1. associate-/r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

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

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

   11. unpow2 N/A

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

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

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

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

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

   14. /-lowering-/.f64 97.7%

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

7. Simplified 97.7%

   \[\leadsto \color{blue}{\frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\frac{1}{x} + \frac{2}{x \cdot x}} + \sqrt[3]{\frac{1}{x}}}} \]

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{2}{3}, x\right), \mathsf{+.f64}\left(\mathsf{cbrt.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}\right), \mathsf{cbrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. /-lowering-/.f64 96.8%

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

10. Simplified 96.8%

    \[\leadsto \frac{\frac{0.6666666666666666}{x}}{\sqrt[3]{\color{blue}{\frac{1}{x}}} + \sqrt[3]{\frac{1}{x}}} \]
11. Add Preprocessing

Alternative 5: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.8%

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

5. Simplified 47.8%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. inv-pow N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. pow-pow N/A

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

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

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

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

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

   9. metadata-eval 96.3%

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

7. Applied egg-rr 96.3%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{-2}} \]
8. Add Preprocessing

Alternative 6: 92.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * (1.0 / 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 * (1.0 / 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 * Float64(1.0 / 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[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $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 9.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.4%

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

   5. Simplified 94.4%

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

      1. pow1/3 N/A

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

      2. inv-pow N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-flip N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

      12. log-pow N/A

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

      13. metadata-eval N/A

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

      14. pow-sqr N/A

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

      15. pow1/3 N/A

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

      16. pow1/3 N/A

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

      17. rec-exp N/A

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

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

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

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

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

      20. pow2 N/A

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

      21. metadata-eval N/A

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

      22. pow-pow N/A

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

      23. rem-cube-cbrt N/A

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

      24. log-pow N/A

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

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

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

      26. log-lowering-log.f64 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. exp-neg N/A

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. pow-to-exp N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. pow1/3 N/A

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

      9. cbrt-div N/A

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

      10. clear-num N/A

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

      11. cbrt-div N/A

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

      12. metadata-eval N/A

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

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

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

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

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

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

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

      16. *-lowering-*.f64 94.5%

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

   9. Applied egg-rr 94.5%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8%

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

   5. Simplified 4.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. rem-cube-cbrt N/A

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

      10. pow-lowering-pow.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

4. Final simplification 91.7%

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

Alternative 7: 92.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{+155}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.8e+155)
   (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x)))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.8e+155) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.8e+155) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) / x));
	} else {
		tmp = 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.8e+155)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) / x)));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.8e+155], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / 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 < 2.80000000000000016e155

   1. Initial program 9.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 93.6%

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

   5. Simplified 93.6%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 94.4%

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

   7. Applied egg-rr 94.4%

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

   if 2.80000000000000016e155 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8%

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

   5. Simplified 4.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. rem-cube-cbrt N/A

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

      10. pow-lowering-pow.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

Alternative 8: 92.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / (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((1.0 / (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(1.0 / 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[(1.0 / N[(x * x), $MachinePrecision]), $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 9.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 94.4%

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

   5. Simplified 94.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.8%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   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 4.8%

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

   5. Simplified 4.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
   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. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-pow N/A

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

      9. rem-cube-cbrt N/A

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

      10. pow-lowering-pow.f64 89.1%

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

   7. Applied egg-rr 89.1%

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

Alternative 9: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \frac{1}{{x}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (/ 1.0 (pow x 0.6666666666666666))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (1.0d0 / (x ** 0.6666666666666666d0))
end function

Java

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

Python

def code(x):
	return 0.3333333333333333 * (1.0 / math.pow(x, 0.6666666666666666))

Julia

function code(x)
	return Float64(0.3333333333333333 * Float64(1.0 / (x ^ 0.6666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (1.0 / (x ^ 0.6666666666666666));
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.8%

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

5. Simplified 47.8%

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

   1. cbrt-div N/A

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

   2. metadata-eval N/A

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

   3. cbrt-prod N/A

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

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

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

   5. pow2 N/A

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

   6. metadata-eval N/A

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

   7. pow-pow N/A

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

   8. rem-cube-cbrt N/A

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

   9. pow-lowering-pow.f64 88.5%

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

7. Applied egg-rr 88.5%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{{x}^{0.6666666666666666}}} \]
8. Add Preprocessing

Alternative 10: 88.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.2%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 47.8%

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

5. Simplified 47.8%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
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 88.5%

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

7. Applied egg-rr 88.5%

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

8. Final simplification 88.5%

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

Alternative 11: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

code[x_] := N[Power[x, 1/3], $MachinePrecision]

Derivation

1. Initial program 7.2%

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

   1. sub-neg N/A

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

   2. sub0-neg N/A

      \[\leadsto 1 + \left(0 - \color{blue}{\sqrt[3]{x}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(0 - \sqrt[3]{x}\right) + \color{blue}{1} \]

7. Applied egg-rr 5.2%

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

8. Taylor expanded in x around inf

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

   1. cbrt-lowering-cbrt.f64 5.2%

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

10. Simplified 5.2%

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

Developer Target 1: 98.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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)))