2cbrt (problem 3.3.4)

Percentage Accurate: 7.2% → 98.5%

Time: 4.8s

Alternatives: 12

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, 0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 2\right) \cdot x, \sqrt[3]{x}, {\left(\sqrt[3]{x - -1}\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  1.0
  (fma
   (*
    (fma (cbrt (pow x -5.0)) 0.3333333333333333 (* (pow (cbrt x) -2.0) 2.0))
    x)
   (cbrt x)
   (pow (cbrt (- x -1.0)) 2.0))))

C

double code(double x) {
	return 1.0 / fma((fma(cbrt(pow(x, -5.0)), 0.3333333333333333, (pow(cbrt(x), -2.0) * 2.0)) * x), cbrt(x), pow(cbrt((x - -1.0)), 2.0));
}

Julia

function code(x)
	return Float64(1.0 / fma(Float64(fma(cbrt((x ^ -5.0)), 0.3333333333333333, Float64((cbrt(x) ^ -2.0) * 2.0)) * x), cbrt(x), (cbrt(Float64(x - -1.0)) ^ 2.0)))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(N[(N[Power[N[Power[x, -5.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333 + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

9. Applied rewrites 98.4%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

12. Applied rewrites 98.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{{x}^{-5}}, 0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{-2} \cdot 2\right) \cdot x}, \sqrt[3]{x}, {\left(\sqrt[3]{x - -1}\right)}^{2}\right)} \]
13. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

9. Applied rewrites 98.4%

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

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x - -1} + \sqrt[3]{x}, \sqrt[3]{x}, {\left(x - -1\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\sqrt[3]{x}\right)}^{-1} \cdot \sqrt[3]{{x}^{-1}}\right) \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+14)
   (/
    1.0
    (fma
     (+ (cbrt (- x -1.0)) (cbrt x))
     (cbrt x)
     (pow (- x -1.0) 0.6666666666666666)))
   (* (* (pow (cbrt x) -1.0) (cbrt (pow x -1.0))) 0.3333333333333333)))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+14) {
		tmp = 1.0 / fma((cbrt((x - -1.0)) + cbrt(x)), cbrt(x), pow((x - -1.0), 0.6666666666666666));
	} else {
		tmp = (pow(cbrt(x), -1.0) * cbrt(pow(x, -1.0))) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.5e+14)
		tmp = Float64(1.0 / fma(Float64(cbrt(Float64(x - -1.0)) + cbrt(x)), cbrt(x), (Float64(x - -1.0) ^ 0.6666666666666666)));
	else
		tmp = Float64(Float64((cbrt(x) ^ -1.0) * cbrt((x ^ -1.0))) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+14], N[(1.0 / N[(N[(N[Power[N[(x - -1.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x - -1.0), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], -1.0], $MachinePrecision] * N[Power[N[Power[x, -1.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e14

   1. Initial program 52.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. rem-cube-cbrt N/A

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

      8. rem-cube-cbrt N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 98.6%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cbrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-cbrt.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-cbrt.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 98.7%

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

       1. lift-pow.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-cbrt.f64 N/A

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

       4. pow1/3 N/A

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

       5. pow-pow N/A

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

       6. metadata-eval N/A

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

       7. lower-pow.f64 N/A

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

       8. lift--.f64 98.2

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

   11. Applied rewrites 98.2%

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

   if 6.5e14 < x

   1. Initial program 4.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 51.9

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 51.9%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.5

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

   7. Applied rewrites 50.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cbrt.f64 N/A

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

      4. inv-pow N/A

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

      5. cbrt-prod N/A

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

      6. unpow-prod-down N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. cbrt-div N/A

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

      10. inv-pow N/A

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

      11. metadata-eval N/A

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

      12. cbrt-div N/A

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

      13. lower-*.f64 N/A

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

      14. cbrt-div N/A

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

      15. metadata-eval N/A

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

      16. inv-pow N/A

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

      17. lower-pow.f64 N/A

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

      18. lift-cbrt.f64 N/A

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

      19. cbrt-div N/A

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

      20. metadata-eval N/A

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

      21. inv-pow N/A

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

      22. lower-pow.f64 N/A

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

      23. lift-cbrt.f64 98.3

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

   9. Applied rewrites 98.3%

      \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333 \]
   10. Step-by-step derivation

       1. lift-cbrt.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow-1 N/A

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

       4. metadata-eval N/A

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

       5. cbrt-div N/A

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

       6. lower-cbrt.f64 N/A

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

       7. inv-pow N/A

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

       8. lower-pow.f64 98.4

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

   11. Applied rewrites 98.4%

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

Alternative 4: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x} \cdot 2, \sqrt[3]{x}, {\left(\sqrt[3]{x - -1}\right)}^{2}\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-cbrt.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-cbrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-cbrt.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

9. Applied rewrites 98.4%

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

10. Taylor expanded in x around inf

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lift-cbrt.f64 96.7

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

12. Applied rewrites 96.7%

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

Alternative 5: 96.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. flip3-- N/A

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

   6. lower-/.f64 N/A

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

   7. rem-cube-cbrt N/A

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

   8. rem-cube-cbrt N/A

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

   9. lower--.f64 N/A

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

   10. metadata-eval N/A

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

   11. fp-cancel-sign-sub-inv N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 N/A

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

   15. lower-+.f64 N/A

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. pow2 N/A

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

   3. lower-*.f64 N/A

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

   4. cbrt-prod N/A

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

   5. pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-cbrt.f64 96.7

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

10. Applied rewrites 96.7%

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

Alternative 6: 96.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 51.4

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

7. Applied rewrites 51.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cbrt.f64 N/A

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

   4. inv-pow N/A

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

   5. cbrt-prod N/A

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

   6. unpow-prod-down N/A

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

   7. inv-pow N/A

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

   8. metadata-eval N/A

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

   9. cbrt-div N/A

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

   10. inv-pow N/A

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

   11. metadata-eval N/A

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

   12. cbrt-div N/A

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

   13. lower-*.f64 N/A

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

   14. cbrt-div N/A

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

   15. metadata-eval N/A

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

   16. inv-pow N/A

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

   17. lower-pow.f64 N/A

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

   18. lift-cbrt.f64 N/A

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

   19. cbrt-div N/A

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

   20. metadata-eval N/A

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

   21. inv-pow N/A

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

   22. lower-pow.f64 N/A

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

   23. lift-cbrt.f64 96.5

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

9. Applied rewrites 96.5%

   \[\leadsto \left({\left(\sqrt[3]{x}\right)}^{-1} \cdot {\left(\sqrt[3]{x}\right)}^{-1}\right) \cdot 0.3333333333333333 \]
10. Step-by-step derivation

    1. lift-cbrt.f64 N/A

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

    2. lift-pow.f64 N/A

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

    3. unpow-1 N/A

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

    4. metadata-eval N/A

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

    5. cbrt-div N/A

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

    6. lower-cbrt.f64 N/A

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

    7. inv-pow N/A

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

    8. lower-pow.f64 96.6

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

11. Applied rewrites 96.6%

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

Alternative 7: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]

6. Taylor expanded in x around inf

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

   1. pow1/3 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-cbrt.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 25.6

       \[\leadsto \frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot \color{blue}{x}} \]

8. Applied rewrites 25.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt[3]{{x}^{4}}, 0.3333333333333333, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{x \cdot x}} \]

9. Taylor expanded in x around inf

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

    1. cbrt-div N/A

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

    2. metadata-eval N/A

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

    3. pow2 N/A

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

    4. *-commutative N/A

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

    5. associate-*l/ N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 N/A

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

    8. cbrt-prod N/A

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

    9. pow2 N/A

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

    10. lower-pow.f64 N/A

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

    11. lift-cbrt.f64 96.5

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

11. Applied rewrites 96.5%

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

Alternative 8: 96.3% accurate, 1.0× speedup

Math

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

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. fp-cancel-sub-sign-inv N/A

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

   6. unpow2 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. lower--.f64 5.9

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

4. Applied rewrites 5.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow1/3 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. pow-pow N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. pow1/3 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-cbrt.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

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

Alternative 9: 91.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 / Math.cbrt((x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / Math.pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 / cbrt(Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / (x ^ 0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.1

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 95.3

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

   7. Applied rewrites 95.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   7. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow2 N/A

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

      4. pow1/3 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.2

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

   9. Applied rewrites 89.2%

      \[\leadsto \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 91.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.cbrt((1.0 / (x * x))) * 0.3333333333333333;
	} else {
		tmp = (1.0 / Math.pow(x, 0.6666666666666666)) * 0.3333333333333333;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(cbrt(Float64(1.0 / Float64(x * x))) * 0.3333333333333333);
	else
		tmp = Float64(Float64(1.0 / (x ^ 0.6666666666666666)) * 0.3333333333333333);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[Power[N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(1.0 / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 95.1

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 95.0

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

   7. Applied rewrites 95.0%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cbrt.f64 N/A

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

      4. pow-flip N/A

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

      5. lower-pow.f64 N/A

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

      6. metadata-eval 7.8

         \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

   5. Applied rewrites 7.8%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
   6. Step-by-step derivation

      1. lift-cbrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. pow-flip N/A

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

      5. cbrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lower-cbrt.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 4.7

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

   7. Applied rewrites 4.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cbrt.f64 N/A

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

      3. pow2 N/A

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

      4. pow1/3 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 89.2

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

   9. Applied rewrites 89.2%

      \[\leadsto \frac{1}{{x}^{0.6666666666666666}} \cdot 0.3333333333333333 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 88.7% accurate, 1.9× speedup

Math

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

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. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cbrt.f64 N/A

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

   4. pow-flip N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval 52.8

      \[\leadsto \sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333 \]

5. Applied rewrites 52.8%

   \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2}} \cdot 0.3333333333333333} \]
6. Step-by-step derivation

   1. lift-cbrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-flip N/A

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

   5. cbrt-div N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-cbrt.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 51.4

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

7. Applied rewrites 51.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cbrt.f64 N/A

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

   5. pow2 N/A

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

   6. cbrt-div N/A

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

   7. pow1/3 N/A

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

   8. pow-flip N/A

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

   9. metadata-eval N/A

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

   10. pow-pow N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 89.0

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

9. Applied rewrites 89.0%

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

Alternative 12: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 1.8%

   \[\leadsto \color{blue}{1} - \sqrt[3]{x} \]
6. Add Preprocessing

Developer Target 1: 98.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025056 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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