2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.5%

Time: 3.9s

Alternatives: 13

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, x, 0.3333333333333333 \cdot {x}^{-0.3333333333333333}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (/
    (+ 1.0 (- x x))
    (fma
     t_0
     t_0
     (fma
      (/ 2.0 (cbrt x))
      x
      (* 0.3333333333333333 (pow x -0.3333333333333333)))))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	return (1.0 + (x - x)) / fma(t_0, t_0, fma((2.0 / cbrt(x)), x, (0.3333333333333333 * pow(x, -0.3333333333333333))));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	return Float64(Float64(1.0 + Float64(x - x)) / fma(t_0, t_0, fma(Float64(2.0 / cbrt(x)), x, Float64(0.3333333333333333 * (x ^ -0.3333333333333333)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(N[(2.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * x + N[(0.3333333333333333 * N[Power[x, -0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-unsound-pow.f64 N/A

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

   11. lower-unsound-fma.f64 N/A

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. lift--.f64 N/A

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

   6. metadata-eval N/A

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

   7. add-flip N/A

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

   8. +-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. rem-cube-cbrt N/A

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

   12. associate--l+ N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 98.5%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \color{blue}{x \cdot \left(\frac{\frac{1}{3}}{{x}^{\frac{4}{3}}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 98.5%

      \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]

8. Applied rewrites 98.5%

   \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \color{blue}{x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)}\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. mult-flip-rev N/A

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

   11. lower-/.f64 N/A

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

   12. lift-cbrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-/.f64 N/A

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

   15. mult-flip N/A

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

   16. associate-*l* N/A

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

   17. lift-pow.f64 N/A

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

   18. pow-flip N/A

       \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, x, \frac{1}{3} \cdot \left({x}^{\left(\mathsf{neg}\left(\frac{4}{3}\right)\right)} \cdot x\right)\right)\right)} \]

   19. pow-plus N/A

       \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, x, \frac{1}{3} \cdot {x}^{\left(\left(\mathsf{neg}\left(\frac{4}{3}\right)\right) + 1\right)}\right)\right)} \]

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval N/A

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

   23. pow-flip N/A

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

   24. pow1/3 N/A

       \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, x, \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]

   25. lift-/.f64 N/A

       \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, x, \frac{1}{3} \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]

10. Applied rewrites 98.5%

    \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \mathsf{fma}\left(\frac{2}{\sqrt[3]{x}}, \color{blue}{x}, 0.3333333333333333 \cdot {x}^{-0.3333333333333333}\right)\right)} \]
11. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt[3]{x - -1}\\ \frac{1}{\mathsf{fma}\left(t\_0, t\_0, x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (- x -1.0))))
   (/
    1.0
    (fma
     t_0
     t_0
     (*
      x
      (+
       (/ 0.3333333333333333 (pow x 1.3333333333333333))
       (* 2.0 (/ 1.0 (cbrt x)))))))))

C

double code(double x) {
	double t_0 = cbrt((x - -1.0));
	return 1.0 / fma(t_0, t_0, (x * ((0.3333333333333333 / pow(x, 1.3333333333333333)) + (2.0 * (1.0 / cbrt(x))))));
}

Julia

function code(x)
	t_0 = cbrt(Float64(x - -1.0))
	return Float64(1.0 / fma(t_0, t_0, Float64(x * Float64(Float64(0.3333333333333333 / (x ^ 1.3333333333333333)) + Float64(2.0 * Float64(1.0 / cbrt(x)))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0 + N[(x * N[(N[(0.3333333333333333 / N[Power[x, 1.3333333333333333], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-unsound-pow.f64 N/A

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

   11. lower-unsound-fma.f64 N/A

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. lift--.f64 N/A

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

   6. metadata-eval N/A

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

   7. add-flip N/A

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

   8. +-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. rem-cube-cbrt N/A

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

   12. associate--l+ N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 98.5%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \color{blue}{x \cdot \left(\frac{\frac{1}{3}}{{x}^{\frac{4}{3}}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)}\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-cbrt.f64 98.5%

      \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]

8. Applied rewrites 98.5%

   \[\leadsto \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, \color{blue}{x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)}\right)} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]
10. Step-by-step derivation

11. Applied rewrites 98.5%

    \[\leadsto \frac{\color{blue}{1}}{\mathsf{fma}\left(\sqrt[3]{x - -1}, \sqrt[3]{x - -1}, x \cdot \left(\frac{0.3333333333333333}{{x}^{1.3333333333333333}} + 2 \cdot \frac{1}{\sqrt[3]{x}}\right)\right)} \]
12. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\left({x}^{0.6666666666666666} - \sqrt[3]{x \cdot \left(-1 - x\right)}\right) - \sqrt[3]{\left(x - -1\right) \cdot \left(-1 - x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+14)
   (/
    (+ 1.0 (- x x))
    (-
     (- (pow x 0.6666666666666666) (cbrt (* x (- -1.0 x))))
     (cbrt (* (- x -1.0) (- -1.0 x)))))
   (/ 0.3333333333333333 (/ 1.0 (pow (cbrt x) -2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+14) {
		tmp = (1.0 + (x - x)) / ((pow(x, 0.6666666666666666) - cbrt((x * (-1.0 - x)))) - cbrt(((x - -1.0) * (-1.0 - x))));
	} else {
		tmp = 0.3333333333333333 / (1.0 / pow(cbrt(x), -2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+14) {
		tmp = (1.0 + (x - x)) / ((Math.pow(x, 0.6666666666666666) - Math.cbrt((x * (-1.0 - x)))) - Math.cbrt(((x - -1.0) * (-1.0 - x))));
	} else {
		tmp = 0.3333333333333333 / (1.0 / Math.pow(Math.cbrt(x), -2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+14)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(Float64((x ^ 0.6666666666666666) - cbrt(Float64(x * Float64(-1.0 - x)))) - cbrt(Float64(Float64(x - -1.0) * Float64(-1.0 - x)))));
	else
		tmp = Float64(0.3333333333333333 / Float64(1.0 / (cbrt(x) ^ -2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+14], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[x, 0.6666666666666666], $MachinePrecision] - N[Power[N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[N[(N[(x - -1.0), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e14

   1. Initial program 7.1%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. add-flip N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-unsound-pow.f64 N/A

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

      11. lower-unsound-fma.f64 N/A

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

   3. Applied rewrites 7.3%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. rem-cube-cbrt N/A

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

      5. lift--.f64 N/A

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

      6. metadata-eval N/A

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

      7. add-flip N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. rem-cube-cbrt N/A

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

      12. associate--l+ N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 98.5%

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 49.8%

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

   if 2e14 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-pow.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. pow-flip N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-unsound-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-flip N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval 88.7%

          \[\leadsto \frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}} \]

   6. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\frac{1}{\color{blue}{{x}^{-0.6666666666666666}}}} \]
   7. Step-by-step derivation

      1. remove-double-div N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. pow-cbrt N/A

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

      7. lift-cbrt.f64 N/A

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

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 96.4%

          \[\leadsto \frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \]

   8. Applied rewrites 96.4%

      \[\leadsto \frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 + (x - x)) / (x * (cbrt(((1.0 / x) + (2.0 * (1.0 / pow(x, 2.0))))) + (2.0 * (1.0 / cbrt(x)))));
}

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. flip3-- N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound--.f64 N/A

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

   5. lower-unsound-pow.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. add-flip N/A

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

   8. lower--.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower-unsound-pow.f64 N/A

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

   11. lower-unsound-fma.f64 N/A

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

3. Applied rewrites 7.3%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. lift--.f64 N/A

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

   6. metadata-eval N/A

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

   7. add-flip N/A

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

   8. +-commutative N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. rem-cube-cbrt N/A

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

   12. associate--l+ N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 98.5%

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-cbrt.f64 97.1%

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

8. Applied rewrites 97.1%

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

Alternative 5: 96.4% accurate, 0.9× speedup

Math

\[\frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 0.3333333333333333 (/ 1.0 (pow (cbrt x) -2.0))))

C

double code(double x) {
	return 0.3333333333333333 / (1.0 / pow(cbrt(x), -2.0));
}

Java

public static double code(double x) {
	return 0.3333333333333333 / (1.0 / Math.pow(Math.cbrt(x), -2.0));
}

Julia

function code(x)
	return Float64(0.3333333333333333 / Float64(1.0 / (cbrt(x) ^ -2.0)))
end

Wolfram

code[x_] := N[(0.3333333333333333 / N[(1.0 / N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7%

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

4. Applied rewrites 88.7%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-neg N/A

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

   8. lower-unsound-pow.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. pow-flip N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-up N/A

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

   13. metadata-eval N/A

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

   14. lift-pow.f64 N/A

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

   15. lower-unsound-/.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. pow-flip N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval 88.7%

       \[\leadsto \frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}} \]

6. Applied rewrites 88.7%

   \[\leadsto \frac{0.3333333333333333}{\frac{1}{\color{blue}{{x}^{-0.6666666666666666}}}} \]
7. Step-by-step derivation

   1. remove-double-div N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-cbrt N/A

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

   7. lift-cbrt.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 96.4%

       \[\leadsto \frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{-2}}} \]

8. Applied rewrites 96.4%

   \[\leadsto \frac{0.3333333333333333}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{-2}}}} \]
9. Add Preprocessing

Alternative 6: 96.4% accurate, 1.0× speedup

Math

\[\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}} \]

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 7.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7%

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

4. Applied rewrites 88.7%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-cbrt N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lower-pow.f64 96.4%

      \[\leadsto \frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{2}}} \]

6. Applied rewrites 96.4%

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

Alternative 7: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7%

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

4. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.7%

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

6. Applied rewrites 88.7%

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

   1. remove-double-div N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. pow-cbrt N/A

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

   7. lift-cbrt.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 96.4%

       \[\leadsto {\left(\sqrt[3]{x}\right)}^{-2} \cdot 0.3333333333333333 \]

8. Applied rewrites 96.4%

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

Alternative 8: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{-0.3333333333333333} \cdot x}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+150)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (/ 0.3333333333333333 (* (pow x -0.3333333333333333) x))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / (pow(x, -0.3333333333333333) * x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / (Math.pow(x, -0.3333333333333333) * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+150)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 / Float64((x ^ -0.3333333333333333) * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+150], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[(N[Power[x, -0.3333333333333333], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999996e150

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 50.0%

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

   6. Applied rewrites 50.0%

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

   if 1.99999999999999996e150 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-unsound-pow.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. pow-flip N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-up N/A

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

      13. metadata-eval N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-unsound-/.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow-flip N/A

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

      18. lower-pow.f64 N/A

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

      19. metadata-eval 88.7%

          \[\leadsto \frac{0.3333333333333333}{\frac{1}{{x}^{-0.6666666666666666}}} \]

   6. Applied rewrites 88.7%

      \[\leadsto \frac{0.3333333333333333}{\frac{1}{\color{blue}{{x}^{-0.6666666666666666}}}} \]
   7. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. pow-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 89.1%

         \[\leadsto \frac{0.3333333333333333}{\frac{1}{e^{\log x \cdot -0.6666666666666666}}} \]

   8. Applied rewrites 89.1%

      \[\leadsto \frac{0.3333333333333333}{\frac{1}{e^{\log x \cdot -0.6666666666666666}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3}}{\frac{1}{\color{blue}{e^{\log x \cdot \frac{-2}{3}}}}} \]

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. exp-to-pow N/A

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

      6. pow-flip N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-plus N/A

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

      11. pow-flip N/A

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

      12. pow1/3 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-cbrt.f64 N/A

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

      15. lower-*.f64 97.0%

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

      16. lift-cbrt.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. pow1/3 N/A

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

      19. pow-flip N/A

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

      20. lower-pow.f64 N/A

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

      21. metadata-eval 90.2%

          \[\leadsto \frac{0.3333333333333333}{{x}^{-0.3333333333333333} \cdot x} \]

   10. Applied rewrites 90.2%

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

Alternative 9: 92.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+150)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (* (exp (* (log x) -0.6666666666666666)) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = exp((log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = Math.exp((Math.log(x) * -0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+150)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(exp(Float64(log(x) * -0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+150], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[x], $MachinePrecision] * -0.6666666666666666), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999996e150

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 50.0%

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

   6. Applied rewrites 50.0%

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

   if 1.99999999999999996e150 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval 88.7%

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

   6. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. rem-exp-log N/A

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

      3. lift-log.f64 N/A

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

      4. pow-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 89.1%

         \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]

   8. Applied rewrites 89.1%

      \[\leadsto e^{\log x \cdot -0.6666666666666666} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{x}^{0.6666666666666666}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e+150)
   (/ 0.3333333333333333 (cbrt (* x x)))
   (/ 0.3333333333333333 (pow x 0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / pow(x, 0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e+150) {
		tmp = 0.3333333333333333 / Math.cbrt((x * x));
	} else {
		tmp = 0.3333333333333333 / Math.pow(x, 0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e+150)
		tmp = Float64(0.3333333333333333 / cbrt(Float64(x * x)));
	else
		tmp = Float64(0.3333333333333333 / (x ^ 0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2e+150], N[(0.3333333333333333 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999996e150

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. pow1/3 N/A

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

      6. lower-cbrt.f64 N/A

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

      7. lower-*.f64 50.0%

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

   6. Applied rewrites 50.0%

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

   if 1.99999999999999996e150 < x

   1. Initial program 7.1%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 88.7%

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

   4. Applied rewrites 88.7%

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

Alternative 11: 88.7% accurate, 1.8× speedup

Math

\[\frac{0.3333333333333333}{{x}^{0.6666666666666666}} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    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 7.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7%

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

4. Applied rewrites 88.7%

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

Alternative 12: 88.7% accurate, 1.9× speedup

Math

\[{x}^{-0.6666666666666666} \cdot 0.3333333333333333 \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-pow.f64 88.7%

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

4. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f64 N/A

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

   8. metadata-eval 88.7%

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

6. Applied rewrites 88.7%

   \[\leadsto {x}^{-0.6666666666666666} \cdot \color{blue}{0.3333333333333333} \]
7. Add Preprocessing

Alternative 13: 4.1% accurate, 36.6× speedup

Math

\[0 \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{0} \]
3. Step-by-step derivation

4. Applied rewrites 4.1%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

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 2025184 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))
  :herbie-expected 5/2

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

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