2cbrt (problem 3.3.4)

Percentage Accurate: 53.0% → 99.6%

Time: 16.8s

Alternatives: 13

Speedup: 1.0×

Specification

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: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := {t_0}^{2}\\ \frac{1}{t_1 + \frac{\sqrt[3]{x}}{\frac{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} - t_0, t_1\right)}{1 + x \cdot 2}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (pow t_0 2.0)))
   (/
    1.0
    (+
     t_1
     (/ (cbrt x) (/ (fma (cbrt x) (- (cbrt x) t_0) t_1) (+ 1.0 (* x 2.0))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = pow(t_0, 2.0);
	return 1.0 / (t_1 + (cbrt(x) / (fma(cbrt(x), (cbrt(x) - t_0), t_1) / (1.0 + (x * 2.0)))));
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = t_0 ^ 2.0
	return Float64(1.0 / Float64(t_1 + Float64(cbrt(x) / Float64(fma(cbrt(x), Float64(cbrt(x) - t_0), t_1) / Float64(1.0 + Float64(x * 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. pow1/3 27.4%

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

4. Applied egg-rr 27.4%

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

   1. pow1/3 55.9%

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

   2. flip3-- 55.9%

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

   3. div-inv 55.9%

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

   4. add-sqr-sqrt 27.2%

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

   5. cbrt-unprod 27.1%

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

   6. rem-cube-cbrt 26.9%

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

   7. cbrt-unprod 27.1%

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

   8. add-sqr-sqrt 55.7%

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

   9. rem-cube-cbrt 56.9%

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

   10. pow2 56.9%

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

   11. distribute-rgt-out 56.9%

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

6. Applied egg-rr 56.9%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

   3. +-commutative 56.9%

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

   4. associate--l+ 99.2%

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

   5. +-inverses 99.2%

      \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   6. metadata-eval 99.2%

      \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   7. +-inverses 99.2%

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

   8. sub-neg 99.2%

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

   9. +-commutative 99.2%

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

   10. *-rgt-identity 99.2%

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

   11. fma-udef 99.2%

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

   12. sub-neg 99.2%

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

   13. fma-udef 99.2%

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

   14. *-rgt-identity 99.2%

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

   15. +-commutative 99.2%

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

   16. sub-neg 99.2%

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

   17. +-inverses 99.2%

       \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   18. metadata-eval 99.2%

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

   19. +-commutative 99.2%

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

8. Simplified 99.2%

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

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. +-commutative 99.2%

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

   5. distribute-rgt-in 99.2%

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

   6. +-commutative 99.2%

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

10. Applied egg-rr 99.2%

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

    1. flip3-+ 99.3%

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

    2. associate-*r/ 86.1%

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

    3. rem-cube-cbrt 86.2%

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

    4. rem-cube-cbrt 86.4%

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

    5. +-commutative 86.4%

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

    6. +-commutative 86.4%

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

    7. pow1/3 69.1%

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

    8. pow1/3 68.8%

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

    9. pow-prod-up 68.8%

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

    10. metadata-eval 68.8%

        \[\leadsto \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(1 + x\right)}^{\color{blue}{0.6666666666666666}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}} \]

    11. metadata-eval 68.8%

        \[\leadsto \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(1 + x\right)}^{\color{blue}{\left(0.16666666666666666 + 0.5\right)}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}} \]

    12. pow-prod-up 69.3%

        \[\leadsto \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{\color{blue}{{\left(1 + x\right)}^{0.16666666666666666} \cdot {\left(1 + x\right)}^{0.5}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}} \]

    13. pow1/2 69.3%

        \[\leadsto \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(1 + x\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt{1 + x}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} - \sqrt[3]{1 + x} \cdot \sqrt[3]{x}\right)}} \]

12. Applied egg-rr 86.4%

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

    1. associate-/l* 99.7%

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

    2. +-commutative 99.7%

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

    3. +-commutative 99.7%

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

    4. associate-+r+ 99.7%

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

    5. +-commutative 99.7%

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

    6. count-2 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 60.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\frac{1}{x + -1}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (* (cbrt (fma x x -1.0)) (cbrt (/ 1.0 (+ x -1.0)))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = (cbrt(fma(x, x, -1.0)) * cbrt((1.0 / (x + -1.0)))) - cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(Float64(cbrt(fma(x, x, -1.0)) * cbrt(Float64(1.0 / Float64(x + -1.0)))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.3%

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

      1. pow1/3 2.7%

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

   4. Applied egg-rr 2.7%

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

      1. pow1/3 4.3%

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

      2. flip3-- 4.3%

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

      3. div-inv 4.3%

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

      4. add-sqr-sqrt 2.3%

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

      5. cbrt-unprod 2.0%

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

      6. rem-cube-cbrt 1.7%

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

      7. cbrt-unprod 1.9%

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

      8. add-sqr-sqrt 3.8%

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

      9. rem-cube-cbrt 4.3%

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

      10. pow2 4.3%

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

      11. distribute-rgt-out 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. associate-*r/ 4.3%

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

      2. *-rgt-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. associate--l+ 98.5%

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

      5. +-inverses 98.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.5%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.5%

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

      8. sub-neg 98.5%

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

      9. +-commutative 98.5%

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

      10. *-rgt-identity 98.5%

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

      11. fma-udef 98.5%

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

      12. sub-neg 98.5%

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

      13. fma-udef 98.5%

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

      14. *-rgt-identity 98.5%

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

      15. +-commutative 98.5%

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

      16. sub-neg 98.5%

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

      17. +-inverses 98.5%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.5%

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

      19. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-rgt-in 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-rgt-in 98.5%

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

      6. +-commutative 98.5%

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

   10. Applied egg-rr 98.5%

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

   11. Taylor expanded in x around 0 19.9%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 98.1%

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

      1. pow1/3 96.5%

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

      2. flip-+ 96.5%

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

      3. div-inv 96.5%

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

      4. unpow-prod-down 1.5%

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

      5. metadata-eval 1.5%

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

      6. fma-neg 1.5%

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

      7. metadata-eval 1.5%

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

      8. sub-neg 1.5%

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

      9. metadata-eval 1.5%

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

   4. Applied egg-rr 1.5%

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

      1. unpow1/3 1.5%

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

      2. unpow1/3 98.1%

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

   6. Simplified 98.1%

      \[\leadsto \color{blue}{\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\frac{1}{x + -1}}} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\frac{1}{x + -1}} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. pow1/3 27.4%

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

4. Applied egg-rr 27.4%

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

   1. pow1/3 55.9%

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

   2. flip3-- 55.9%

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

   3. div-inv 55.9%

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

   4. add-sqr-sqrt 27.2%

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

   5. cbrt-unprod 27.1%

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

   6. rem-cube-cbrt 26.9%

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

   7. cbrt-unprod 27.1%

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

   8. add-sqr-sqrt 55.7%

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

   9. rem-cube-cbrt 56.9%

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

   10. pow2 56.9%

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

   11. distribute-rgt-out 56.9%

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

6. Applied egg-rr 56.9%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

   3. +-commutative 56.9%

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

   4. associate--l+ 99.2%

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

   5. +-inverses 99.2%

      \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   6. metadata-eval 99.2%

      \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   7. +-inverses 99.2%

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

   8. sub-neg 99.2%

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

   9. +-commutative 99.2%

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

   10. *-rgt-identity 99.2%

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

   11. fma-udef 99.2%

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

   12. sub-neg 99.2%

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

   13. fma-udef 99.2%

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

   14. *-rgt-identity 99.2%

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

   15. +-commutative 99.2%

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

   16. sub-neg 99.2%

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

   17. +-inverses 99.2%

       \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   18. metadata-eval 99.2%

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

   19. +-commutative 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 4: 88.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 t_0))
     (if (<= x 1.32e+154)
       (/ 1.0 (+ t_0 (cbrt (pow (+ 1.0 x) 2.0))))
       (/ 1.0 (+ t_0 (exp (* 0.6666666666666666 (log1p x)))))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / (t_0 + cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + t_0);
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / (t_0 + Math.cbrt(Math.pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / (t_0 + Math.exp((0.6666666666666666 * Math.log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	elseif (x <= 1.32e+154)
		tmp = Float64(1.0 / Float64(t_0 + cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(1.0 / Float64(t_0 + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(1.0 / N[(t$95$0 + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.8%

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

      1. pow1/3 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. pow1/3 4.8%

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

      2. flip3-- 4.8%

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

      3. div-inv 4.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. cbrt-unprod 0.0%

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

      6. rem-cube-cbrt 0.0%

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

      7. cbrt-unprod 0.0%

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

      8. add-sqr-sqrt 3.6%

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

      9. rem-cube-cbrt 4.8%

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

      10. pow2 4.8%

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

      11. distribute-rgt-out 4.8%

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

   6. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 98.3%

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

      5. +-inverses 98.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.3%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.3%

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

      8. sub-neg 98.3%

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

      9. +-commutative 98.3%

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

      10. *-rgt-identity 98.3%

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

      11. fma-udef 98.3%

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

      12. sub-neg 98.3%

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

      13. fma-udef 98.3%

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

      14. *-rgt-identity 98.3%

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

      15. +-commutative 98.3%

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

      16. sub-neg 98.3%

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

      17. +-inverses 98.3%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.3%

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

      19. +-commutative 98.3%

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

   8. Simplified 98.4%

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

      1. fma-udef 98.3%

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

      2. +-commutative 98.3%

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

      3. distribute-rgt-in 98.3%

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

      4. +-commutative 98.3%

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

      5. distribute-rgt-in 98.3%

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

      6. +-commutative 98.3%

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

   10. Applied egg-rr 98.3%

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

   11. Taylor expanded in x around 0 20.0%

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

   if -1.35000000000000003e154 < x < 1.31999999999999998e154

   1. Initial program 74.1%

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

      1. pow1/3 36.5%

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

   4. Applied egg-rr 36.5%

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

      1. pow1/3 74.1%

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

      2. flip3-- 74.0%

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

      3. div-inv 74.0%

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

      4. add-sqr-sqrt 36.1%

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

      5. cbrt-unprod 36.1%

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

      6. rem-cube-cbrt 36.0%

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

      7. cbrt-unprod 36.1%

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

      8. add-sqr-sqrt 74.3%

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

      9. rem-cube-cbrt 75.4%

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

      10. pow2 75.4%

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

      11. distribute-rgt-out 75.4%

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

   6. Applied egg-rr 75.4%

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

      1. associate-*r/ 75.4%

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

      2. *-rgt-identity 75.4%

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

      3. +-commutative 75.4%

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

      4. associate--l+ 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. fma-udef 99.5%

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

      12. sub-neg 99.5%

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

      13. fma-udef 99.5%

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

      14. *-rgt-identity 99.5%

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

      15. +-commutative 99.5%

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

      16. sub-neg 99.5%

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

      17. +-inverses 99.5%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-rgt-in 99.5%

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

      6. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. unpow2 99.5%

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

       2. cbrt-unprod 99.6%

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

       3. add-exp-log 86.6%

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

       4. log1p-udef 86.6%

          \[\leadsto \frac{1}{\sqrt[3]{e^{\color{blue}{\mathsf{log1p}\left(x\right)}} \cdot \left(1 + x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       5. add-exp-log 86.4%

          \[\leadsto \frac{1}{\sqrt[3]{e^{\mathsf{log1p}\left(x\right)} \cdot \color{blue}{e^{\log \left(1 + x\right)}}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       6. log1p-udef 86.4%

          \[\leadsto \frac{1}{\sqrt[3]{e^{\mathsf{log1p}\left(x\right)} \cdot e^{\color{blue}{\mathsf{log1p}\left(x\right)}}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       7. log1p-udef 86.4%

          \[\leadsto \frac{1}{\sqrt[3]{e^{\color{blue}{\log \left(1 + x\right)}} \cdot e^{\mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       8. add-exp-log 86.6%

          \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{\left(1 + x\right)} \cdot e^{\mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       9. log1p-udef 86.6%

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

       10. add-exp-log 99.6%

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

       11. pow2 99.6%

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

   12. Applied egg-rr 99.6%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.8%

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

      1. pow1/3 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. pow1/3 4.8%

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

      2. flip3-- 4.8%

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

      3. div-inv 4.8%

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

      4. add-sqr-sqrt 4.7%

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

      5. cbrt-unprod 3.6%

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

      6. rem-cube-cbrt 2.9%

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

      7. cbrt-unprod 3.5%

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

      8. add-sqr-sqrt 3.1%

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

      9. rem-cube-cbrt 4.8%

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

      10. pow2 4.8%

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

      11. distribute-rgt-out 4.8%

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

   6. Applied egg-rr 4.8%

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

      1. associate-*r/ 4.8%

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

      2. *-rgt-identity 4.8%

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

      3. +-commutative 4.8%

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

      4. associate--l+ 98.7%

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

      5. +-inverses 98.7%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.7%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.7%

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

      8. sub-neg 98.7%

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

      9. +-commutative 98.7%

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

      10. *-rgt-identity 98.7%

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

      11. fma-udef 98.7%

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

      12. sub-neg 98.7%

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

      13. fma-udef 98.7%

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

      14. *-rgt-identity 98.7%

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

      15. +-commutative 98.7%

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

      16. sub-neg 98.7%

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

      17. +-inverses 98.7%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.7%

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

      19. +-commutative 98.7%

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

   8. Simplified 98.7%

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

      1. fma-udef 98.7%

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

      2. +-commutative 98.7%

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

      3. distribute-rgt-in 98.7%

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

      4. +-commutative 98.7%

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

      5. distribute-rgt-in 98.7%

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

      6. +-commutative 98.7%

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

   10. Applied egg-rr 98.7%

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

       1. add-exp-log 93.1%

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

       2. log-pow 93.1%

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

       3. pow1/3 92.5%

          \[\leadsto \frac{1}{e^{2 \cdot \log \color{blue}{\left({\left(1 + x\right)}^{0.3333333333333333}\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       4. log-pow 92.6%

          \[\leadsto \frac{1}{e^{2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(1 + x\right)\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       5. log1p-udef 92.6%

          \[\leadsto \frac{1}{e^{2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   12. Applied egg-rr 92.6%

       \[\leadsto \frac{1}{\color{blue}{e^{2 \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 92.6%

          \[\leadsto \frac{1}{e^{\color{blue}{\left(2 \cdot 0.3333333333333333\right) \cdot \mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       2. metadata-eval 92.6%

          \[\leadsto \frac{1}{e^{\color{blue}{0.6666666666666666} \cdot \mathsf{log1p}\left(x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   14. Simplified 92.6%

       \[\leadsto \frac{1}{\color{blue}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1 - {x}^{2}}{1 - x}} - \sqrt[3]{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ t_0 (cbrt x)))))
     (- (cbrt (/ (- 1.0 (pow x 2.0)) (- 1.0 x))) (cbrt x)))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (t_0 + cbrt(x))));
	} else {
		tmp = cbrt(((1.0 - pow(x, 2.0)) / (1.0 - x))) - cbrt(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.cbrt(x)) <= 0.0) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
	} else {
		tmp = Math.cbrt(((1.0 - Math.pow(x, 2.0)) / (1.0 - x))) - Math.cbrt(x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(t_0 + cbrt(x)))));
	else
		tmp = Float64(cbrt(Float64(Float64(1.0 - (x ^ 2.0)) / Float64(1.0 - x))) - cbrt(x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(t$95$0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[(1.0 - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

   1. Initial program 4.3%

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

      1. pow1/3 2.7%

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

   4. Applied egg-rr 2.7%

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

      1. pow1/3 4.3%

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

      2. flip3-- 4.3%

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

      3. div-inv 4.3%

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

      4. add-sqr-sqrt 2.3%

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

      5. cbrt-unprod 2.0%

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

      6. rem-cube-cbrt 1.7%

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

      7. cbrt-unprod 1.9%

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

      8. add-sqr-sqrt 3.8%

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

      9. rem-cube-cbrt 4.3%

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

      10. pow2 4.3%

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

      11. distribute-rgt-out 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. associate-*r/ 4.3%

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

      2. *-rgt-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. associate--l+ 98.5%

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

      5. +-inverses 98.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.5%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.5%

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

      8. sub-neg 98.5%

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

      9. +-commutative 98.5%

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

      10. *-rgt-identity 98.5%

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

      11. fma-udef 98.5%

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

      12. sub-neg 98.5%

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

      13. fma-udef 98.5%

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

      14. *-rgt-identity 98.5%

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

      15. +-commutative 98.5%

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

      16. sub-neg 98.5%

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

      17. +-inverses 98.5%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.5%

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

      19. +-commutative 98.5%

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

   8. Simplified 98.5%

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

      1. fma-udef 98.5%

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

      2. +-commutative 98.5%

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

      3. distribute-rgt-in 98.5%

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

      4. +-commutative 98.5%

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

      5. distribute-rgt-in 98.5%

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

      6. +-commutative 98.5%

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

   10. Applied egg-rr 98.5%

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

   11. Taylor expanded in x around 0 19.9%

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

   if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

   1. Initial program 98.1%

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

      1. add-exp-log 96.5%

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

      2. pow1/3 96.6%

         \[\leadsto e^{\log \color{blue}{\left({\left(x + 1\right)}^{0.3333333333333333}\right)}} - \sqrt[3]{x} \]

      3. log-pow 96.5%

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

      4. +-commutative 96.5%

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

      5. log1p-udef 96.5%

         \[\leadsto e^{0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]

   4. Applied egg-rr 96.5%

      \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]
   5. Step-by-step derivation

      1. *-commutative 96.5%

         \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 0.3333333333333333}} - \sqrt[3]{x} \]

      2. exp-prod 96.5%

         \[\leadsto \color{blue}{{\left(e^{\mathsf{log1p}\left(x\right)}\right)}^{0.3333333333333333}} - \sqrt[3]{x} \]

      3. unpow1/3 96.5%

         \[\leadsto \color{blue}{\sqrt[3]{e^{\mathsf{log1p}\left(x\right)}}} - \sqrt[3]{x} \]

   6. Simplified 96.5%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\mathsf{log1p}\left(x\right)}}} - \sqrt[3]{x} \]
   7. Step-by-step derivation

      1. log1p-udef 96.5%

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

      2. add-exp-log 98.1%

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

      3. flip-+ 98.1%

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

      4. metadata-eval 98.1%

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

      5. div-sub 98.1%

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

      6. pow2 98.1%

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

   8. Applied egg-rr 98.1%

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

      1. div-sub 98.1%

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

   10. Simplified 98.1%

       \[\leadsto \sqrt[3]{\color{blue}{\frac{1 - {x}^{2}}{1 - x}}} - \sqrt[3]{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1 - {x}^{2}}{1 - x}} - \sqrt[3]{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (cbrt x) (+ (cbrt (+ 1.0 x)) (cbrt x)))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (+ t_0 (exp (* 0.6666666666666666 (log1p x))))))))

C

double code(double x) {
	double t_0 = cbrt(x) * (cbrt((1.0 + x)) + cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt(x) * (Math.cbrt((1.0 + x)) + Math.cbrt(x));
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + t_0);
	} else {
		tmp = 1.0 / (t_0 + Math.exp((0.6666666666666666 * Math.log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(x) * Float64(cbrt(Float64(1.0 + x)) + cbrt(x)))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + t_0));
	else
		tmp = Float64(1.0 / Float64(t_0 + exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 7.7%

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

      1. pow1/3 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. pow1/3 7.7%

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

      2. flip3-- 7.7%

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

      3. div-inv 7.7%

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

      4. add-sqr-sqrt 0.0%

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

      5. cbrt-unprod 0.0%

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

      6. rem-cube-cbrt 0.0%

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

      7. cbrt-unprod 0.0%

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

      8. add-sqr-sqrt 7.9%

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

      9. rem-cube-cbrt 9.1%

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

      10. pow2 9.1%

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

      11. distribute-rgt-out 9.1%

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

   6. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate--l+ 98.4%

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

      5. +-inverses 98.4%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.4%

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

      8. sub-neg 98.4%

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

      9. +-commutative 98.4%

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

      10. *-rgt-identity 98.4%

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

      11. fma-udef 98.4%

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

      12. sub-neg 98.4%

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

      13. fma-udef 98.4%

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

      14. *-rgt-identity 98.4%

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

      15. +-commutative 98.4%

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

      16. sub-neg 98.4%

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

      17. +-inverses 98.4%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.4%

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

      19. +-commutative 98.4%

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

   8. Simplified 98.5%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. distribute-rgt-in 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-rgt-in 98.4%

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

      6. +-commutative 98.4%

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

   10. Applied egg-rr 98.4%

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

   11. Taylor expanded in x around 0 20.0%

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

   if -1 < x

   1. Initial program 70.7%

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

      1. pow1/3 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. pow1/3 70.7%

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

      2. flip3-- 70.7%

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

      3. div-inv 70.7%

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

      4. add-sqr-sqrt 35.6%

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

      5. cbrt-unprod 35.3%

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

      6. rem-cube-cbrt 35.2%

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

      7. cbrt-unprod 35.4%

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

      8. add-sqr-sqrt 70.4%

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

      9. rem-cube-cbrt 71.5%

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

      10. pow2 71.5%

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

      11. distribute-rgt-out 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*r/ 71.5%

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

      2. *-rgt-identity 71.5%

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

      3. +-commutative 71.5%

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

      4. associate--l+ 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. fma-udef 99.5%

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

      12. sub-neg 99.5%

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

      13. fma-udef 99.5%

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

      14. *-rgt-identity 99.5%

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

      15. +-commutative 99.5%

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

      16. sub-neg 99.5%

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

      17. +-inverses 99.5%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-rgt-in 99.5%

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

      6. +-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

       1. add-exp-log 98.1%

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

       2. log-pow 98.1%

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

       3. pow1/3 97.9%

          \[\leadsto \frac{1}{e^{2 \cdot \log \color{blue}{\left({\left(1 + x\right)}^{0.3333333333333333}\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       4. log-pow 97.9%

          \[\leadsto \frac{1}{e^{2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(1 + x\right)\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       5. log1p-udef 97.9%

          \[\leadsto \frac{1}{e^{2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   12. Applied egg-rr 97.9%

       \[\leadsto \frac{1}{\color{blue}{e^{2 \cdot \left(0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 97.9%

          \[\leadsto \frac{1}{e^{\color{blue}{\left(2 \cdot 0.3333333333333333\right) \cdot \mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

       2. metadata-eval 97.9%

          \[\leadsto \frac{1}{e^{\color{blue}{0.6666666666666666} \cdot \mathsf{log1p}\left(x\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

   14. Simplified 97.9%

       \[\leadsto \frac{1}{\color{blue}{e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x} + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (cbrt (+ 1.0 x)) (cbrt x))))
   (if (<= x -1.0)
     (/ 1.0 (+ 1.0 (* (cbrt x) t_0)))
     (/ 1.0 (fma (cbrt x) t_0 (pow (+ 1.0 x) 0.6666666666666666))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x)) + cbrt(x);
	double tmp;
	if (x <= -1.0) {
		tmp = 1.0 / (1.0 + (cbrt(x) * t_0));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_0, pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(cbrt(Float64(1.0 + x)) + cbrt(x))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * t_0)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_0, (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 7.7%

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

      1. pow1/3 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. pow1/3 7.7%

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

      2. flip3-- 7.7%

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

      3. div-inv 7.7%

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

      4. add-sqr-sqrt 0.0%

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

      5. cbrt-unprod 0.0%

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

      6. rem-cube-cbrt 0.0%

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

      7. cbrt-unprod 0.0%

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

      8. add-sqr-sqrt 7.9%

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

      9. rem-cube-cbrt 9.1%

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

      10. pow2 9.1%

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

      11. distribute-rgt-out 9.1%

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

   6. Applied egg-rr 9.1%

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

      1. associate-*r/ 9.1%

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

      2. *-rgt-identity 9.1%

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

      3. +-commutative 9.1%

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

      4. associate--l+ 98.4%

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

      5. +-inverses 98.4%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 98.4%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 98.4%

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

      8. sub-neg 98.4%

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

      9. +-commutative 98.4%

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

      10. *-rgt-identity 98.4%

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

      11. fma-udef 98.4%

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

      12. sub-neg 98.4%

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

      13. fma-udef 98.4%

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

      14. *-rgt-identity 98.4%

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

      15. +-commutative 98.4%

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

      16. sub-neg 98.4%

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

      17. +-inverses 98.4%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 98.4%

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

      19. +-commutative 98.4%

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

   8. Simplified 98.5%

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

      1. fma-udef 98.4%

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

      2. +-commutative 98.4%

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

      3. distribute-rgt-in 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-rgt-in 98.4%

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

      6. +-commutative 98.4%

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

   10. Applied egg-rr 98.4%

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

   11. Taylor expanded in x around 0 20.0%

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

   if -1 < x

   1. Initial program 70.7%

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

      1. pow1/3 35.8%

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

   4. Applied egg-rr 35.8%

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

      1. pow1/3 70.7%

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

      2. flip3-- 70.7%

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

      3. div-inv 70.7%

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

      4. add-sqr-sqrt 35.6%

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

      5. cbrt-unprod 35.3%

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

      6. rem-cube-cbrt 35.2%

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

      7. cbrt-unprod 35.4%

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

      8. add-sqr-sqrt 70.4%

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

      9. rem-cube-cbrt 71.5%

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

      10. pow2 71.5%

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

      11. distribute-rgt-out 71.5%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*r/ 71.5%

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

      2. *-rgt-identity 71.5%

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

      3. +-commutative 71.5%

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

      4. associate--l+ 99.5%

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

      5. +-inverses 99.5%

         \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      7. +-inverses 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. *-rgt-identity 99.5%

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

      11. fma-udef 99.5%

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

      12. sub-neg 99.5%

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

      13. fma-udef 99.5%

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

      14. *-rgt-identity 99.5%

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

      15. +-commutative 99.5%

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

      16. sub-neg 99.5%

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

      17. +-inverses 99.5%

          \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

      18. metadata-eval 99.5%

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

      19. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. unpow2 99.5%

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

      2. add-sqr-sqrt 99.4%

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

      3. associate-*r* 99.4%

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

      4. add-sqr-sqrt 99.4%

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

      5. sqrt-prod 99.4%

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

      6. unpow2 99.4%

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

      7. sqrt-prod 99.4%

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

      8. unpow2 99.4%

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

      9. add-cube-cbrt 99.6%

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

      10. add-exp-log 98.2%

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

      11. log1p-udef 98.2%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt{e^{\color{blue}{\mathsf{log1p}\left(x\right)}}} \cdot \sqrt{\sqrt[3]{1 + x}}\right)} \]

      12. log1p-udef 98.2%

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

      13. add-exp-log 99.6%

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

      14. add-sqr-sqrt 64.4%

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

      15. hypot-1-def 64.4%

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

      16. pow1/3 63.4%

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

      17. sqrt-pow1 63.4%

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

   10. Applied egg-rr 63.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right) \cdot {\left(1 + x\right)}^{0.16666666666666666}}\right)} \]
   11. Step-by-step derivation

       1. *-commutative 63.4%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(1 + x\right)}^{0.16666666666666666} \cdot \mathsf{hypot}\left(1, \sqrt{x}\right)}\right)} \]

       2. hypot-1-def 63.4%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt{1 + \sqrt{x} \cdot \sqrt{x}}}\right)} \]

       3. rem-square-sqrt 98.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.16666666666666666} \cdot \sqrt{1 + \color{blue}{x}}\right)} \]

   12. Simplified 98.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(1 + x\right)}^{0.16666666666666666} \cdot \sqrt{1 + x}}\right)} \]
   13. Step-by-step derivation

       1. pow1/2 98.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(1 + x\right)}^{0.5}}\right)} \]

       2. pow-prod-up 97.7%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(1 + x\right)}^{\left(0.16666666666666666 + 0.5\right)}}\right)} \]

       3. +-commutative 97.7%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(x + 1\right)}}^{\left(0.16666666666666666 + 0.5\right)}\right)} \]

       4. metadata-eval 97.7%

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

   14. Applied egg-rr 97.7%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(x + 1\right)}^{0.6666666666666666}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	return 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (t_0 + Math.cbrt(x))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. pow1/3 27.4%

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

4. Applied egg-rr 27.4%

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

   1. pow1/3 55.9%

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

   2. flip3-- 55.9%

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

   3. div-inv 55.9%

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

   4. add-sqr-sqrt 27.2%

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

   5. cbrt-unprod 27.1%

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

   6. rem-cube-cbrt 26.9%

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

   7. cbrt-unprod 27.1%

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

   8. add-sqr-sqrt 55.7%

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

   9. rem-cube-cbrt 56.9%

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

   10. pow2 56.9%

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

   11. distribute-rgt-out 56.9%

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

6. Applied egg-rr 56.9%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

   3. +-commutative 56.9%

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

   4. associate--l+ 99.2%

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

   5. +-inverses 99.2%

      \[\leadsto \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   6. metadata-eval 99.2%

      \[\leadsto \frac{1 + \color{blue}{\left(-0\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   7. +-inverses 99.2%

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

   8. sub-neg 99.2%

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

   9. +-commutative 99.2%

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

   10. *-rgt-identity 99.2%

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

   11. fma-udef 99.2%

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

   12. sub-neg 99.2%

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

   13. fma-udef 99.2%

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

   14. *-rgt-identity 99.2%

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

   15. +-commutative 99.2%

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

   16. sub-neg 99.2%

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

   17. +-inverses 99.2%

       \[\leadsto \frac{1 - \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]

   18. metadata-eval 99.2%

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

   19. +-commutative 99.2%

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

8. Simplified 99.2%

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

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-rgt-in 99.2%

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

   4. +-commutative 99.2%

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

   5. distribute-rgt-in 99.2%

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

   6. +-commutative 99.2%

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

10. Applied egg-rr 99.2%

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

11. Final simplification 99.2%

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

Alternative 9: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

3. Final simplification 55.9%

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

Alternative 10: 50.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.9%

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

   1. add-cube-cbrt 55.7%

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

   2. pow3 55.7%

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

4. Applied egg-rr 55.7%

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

5. Taylor expanded in x around 0 26.2%

   \[\leadsto \color{blue}{\left(1 + 0.3333333333333333 \cdot x\right) - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. associate--l+ 26.2%

      \[\leadsto \color{blue}{1 + \left(0.3333333333333333 \cdot x - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right)} \]

   2. *-commutative 26.2%

      \[\leadsto 1 + \left(\color{blue}{x \cdot 0.3333333333333333} - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   3. metadata-eval 26.2%

      \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}\right) \]

   4. pow-base-1 26.2%

      \[\leadsto 1 + \left(x \cdot 0.3333333333333333 - \color{blue}{1} \cdot {x}^{0.3333333333333333}\right) \]

   5. unpow1/3 53.3%

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

   6. *-lft-identity 53.3%

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

7. Simplified 53.3%

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

8. Final simplification 53.3%

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

Alternative 11: 50.4% 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 55.9%

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

   1. add-cube-cbrt 55.7%

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

   2. pow3 55.7%

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

4. Applied egg-rr 55.7%

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

5. Taylor expanded in x around 0 24.9%

   \[\leadsto \color{blue}{1 - {\left({1}^{4}\right)}^{0.1111111111111111} \cdot {x}^{0.3333333333333333}} \]
6. Step-by-step derivation

   1. metadata-eval 24.9%

      \[\leadsto 1 - {\color{blue}{1}}^{0.1111111111111111} \cdot {x}^{0.3333333333333333} \]

   2. pow-base-1 24.9%

      \[\leadsto 1 - \color{blue}{1} \cdot {x}^{0.3333333333333333} \]

   3. unpow1/3 52.4%

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

   4. *-lft-identity 52.4%

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

7. Simplified 52.4%

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

8. Final simplification 52.4%

   \[\leadsto 1 - \sqrt[3]{x} \]
9. Add Preprocessing

Alternative 12: 3.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    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 55.9%

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

3. Taylor expanded in x around inf 3.6%

   \[\leadsto \color{blue}{0} \]

4. Final simplification 3.6%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 13: 49.7% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 55.9%

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

3. Taylor expanded in x around 0 51.0%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 51.0%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))