2cbrt (problem 3.3.4)

Percentage Accurate: 7.3% → 98.6%

Time: 18.5s

Alternatives: 14

Speedup: 1.9×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{x}}\\ t_1 := \sqrt[3]{1 + x}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t\_0 \cdot t\_0 + t\_1, {t\_1}^{2}\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. flip3-- 7.0%

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

   2. rem-cube-cbrt 6.5%

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

   3. rem-cube-cbrt 9.0%

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

   4. div-sub 7.1%

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

   5. cbrt-unprod 5.9%

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

   6. distribute-rgt-out 5.9%

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

   7. +-commutative 5.9%

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

4. Applied egg-rr 7.1%

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

   1. div-sub 9.0%

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

   2. +-commutative 9.0%

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

   3. associate--l+ 53.2%

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

   4. +-inverses 53.2%

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

   5. metadata-eval 53.2%

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

   6. +-commutative 53.2%

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

   7. fma-define 53.2%

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

   8. +-commutative 53.2%

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

   9. +-commutative 53.2%

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

   10. rem-exp-log 52.1%

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

   11. +-commutative 52.1%

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

   12. log1p-undefine 52.1%

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

   13. rem-exp-log 51.5%

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

   14. +-commutative 51.5%

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

   15. log1p-undefine 51.5%

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

6. Simplified 51.5%

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

   1. pow1/3 51.1%

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

   2. exp-sum 51.1%

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

   3. log1p-undefine 51.1%

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

   4. add-exp-log 51.1%

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

   5. log1p-undefine 51.1%

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

   6. add-exp-log 51.0%

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

   7. unpow-prod-down 93.1%

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

   8. +-commutative 93.1%

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

   9. pow1/3 94.5%

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

   10. +-commutative 94.5%

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

   11. pow1/3 98.5%

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

   12. pow2 98.5%

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

8. Applied egg-rr 98.5%

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

   1. add-cube-cbrt 98.3%

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

   2. pow3 98.4%

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

10. Applied egg-rr 98.4%

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

    1. rem-cube-cbrt 98.5%

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

    2. pow1/3 94.5%

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

    3. add-sqr-sqrt 94.5%

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

    4. unpow-prod-down 94.5%

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

12. Applied egg-rr 94.5%

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

    1. unpow1/3 95.9%

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

    2. unpow1/3 98.5%

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

14. Simplified 98.5%

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

15. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), (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[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.9%

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

   1. flip3-- 7.0%

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

   2. rem-cube-cbrt 6.5%

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

   3. rem-cube-cbrt 9.0%

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

   4. div-sub 7.1%

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

   5. cbrt-unprod 5.9%

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

   6. distribute-rgt-out 5.9%

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

   7. +-commutative 5.9%

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

4. Applied egg-rr 7.1%

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

   1. div-sub 9.0%

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

   2. +-commutative 9.0%

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

   3. associate--l+ 53.2%

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

   4. +-inverses 53.2%

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

   5. metadata-eval 53.2%

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

   6. +-commutative 53.2%

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

   7. fma-define 53.2%

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

   8. +-commutative 53.2%

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

   9. +-commutative 53.2%

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

   10. rem-exp-log 52.1%

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

   11. +-commutative 52.1%

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

   12. log1p-undefine 52.1%

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

   13. rem-exp-log 51.5%

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

   14. +-commutative 51.5%

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

   15. log1p-undefine 51.5%

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

6. Simplified 51.5%

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

   1. pow1/3 51.1%

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

   2. exp-sum 51.1%

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

   3. log1p-undefine 51.1%

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

   4. add-exp-log 51.1%

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

   5. log1p-undefine 51.1%

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

   6. add-exp-log 51.0%

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

   7. unpow-prod-down 93.1%

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

   8. +-commutative 93.1%

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

   9. pow1/3 94.5%

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

   10. +-commutative 94.5%

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

   11. pow1/3 98.5%

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

   12. pow2 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 97.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

   1. flip3-- 7.0%

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

   2. rem-cube-cbrt 6.5%

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

   3. rem-cube-cbrt 9.0%

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

   4. div-sub 7.1%

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

   5. cbrt-unprod 5.9%

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

   6. distribute-rgt-out 5.9%

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

   7. +-commutative 5.9%

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

4. Applied egg-rr 7.1%

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

   1. div-sub 9.0%

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

   2. +-commutative 9.0%

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

   3. associate--l+ 53.2%

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

   4. +-inverses 53.2%

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

   5. metadata-eval 53.2%

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

   6. +-commutative 53.2%

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

   7. fma-define 53.2%

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

   8. +-commutative 53.2%

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

   9. +-commutative 53.2%

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

   10. rem-exp-log 52.1%

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

   11. +-commutative 52.1%

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

   12. log1p-undefine 52.1%

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

   13. rem-exp-log 51.5%

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

   14. +-commutative 51.5%

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

   15. log1p-undefine 51.5%

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

6. Simplified 51.5%

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

   1. pow1/3 51.1%

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

   2. exp-sum 51.1%

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

   3. log1p-undefine 51.1%

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

   4. add-exp-log 51.1%

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

   5. log1p-undefine 51.1%

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

   6. add-exp-log 51.0%

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

   7. unpow-prod-down 93.1%

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

   8. +-commutative 93.1%

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

   9. pow1/3 94.5%

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

   10. +-commutative 94.5%

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

   11. pow1/3 98.5%

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

   12. pow2 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around inf 97.8%

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

    1. unpow2 97.8%

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

    2. associate-*r/ 97.8%

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

    3. metadata-eval 97.8%

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

11. Simplified 97.8%

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

Alternative 4: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (*
    (/ 1.0 x)
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (/ 0.06172839506172839 (cbrt (* x x)))
       (* 0.3333333333333333 (* x (sqrt (* x (/ 1.0 (cbrt x))))))))
     x))
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ 1.0 x)))
     (pow (+ 1.0 x) 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (1.0 / x) * (((cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / cbrt((x * x))) + (0.3333333333333333 * (x * sqrt((x * (1.0 / cbrt(x)))))))) / x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.06172839506172839 / cbrt(Float64(x * x))) + Float64(0.3333333333333333 * Float64(x * sqrt(Float64(x * Float64(1.0 / cbrt(x)))))))) / x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.06172839506172839 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Sqrt[N[(x * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf 32.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. sqrt-unprod 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. sqr-neg 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt{\color{blue}{\left(-\sqrt[3]{x \cdot x}\right) \cdot \left(-\sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{-\sqrt[3]{x \cdot x}} \cdot \sqrt{-\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(-\sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      6. neg-sub0 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(0 - \sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      7. flip3-- 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt[3]{x \cdot x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   7. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{x \cdot x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. +-lft-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. mul0-lft 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{x \cdot \sqrt[3]{x} + \color{blue}{0}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. +-rgt-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   9. Simplified 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{x \cdot \sqrt[3]{x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 5.1%

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

      4. div-sub 5.1%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 5.1%

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

      1. div-sub 5.1%

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

      2. +-commutative 5.1%

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

      3. associate--l+ 5.1%

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

      4. +-inverses 5.1%

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

      5. metadata-eval 5.1%

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

      6. +-commutative 5.1%

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

      7. fma-define 5.1%

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

      8. +-commutative 5.1%

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

      9. +-commutative 5.1%

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

      10. rem-exp-log 5.1%

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

      11. +-commutative 5.1%

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

      12. log1p-undefine 5.1%

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

      13. rem-exp-log 5.1%

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

      14. +-commutative 5.1%

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

      15. log1p-undefine 5.1%

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

   6. Simplified 5.1%

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

      1. pow1/3 5.1%

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

      2. exp-sum 5.1%

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

      3. log1p-undefine 5.1%

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

      4. add-exp-log 5.1%

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

      5. log1p-undefine 5.1%

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

      6. add-exp-log 5.1%

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

      7. unpow-prod-down 91.3%

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

      8. pow-prod-up 91.3%

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

      9. +-commutative 91.3%

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

      10. metadata-eval 91.3%

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

   8. Applied egg-rr 91.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (*
    (/ 1.0 x)
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (/ 0.06172839506172839 (cbrt (* x x)))
       (* 0.3333333333333333 (* x (sqrt (* x (/ 1.0 (cbrt x))))))))
     x))
   (/
    1.0
    (fma (cbrt x) (+ (cbrt x) (cbrt x)) (pow (+ 1.0 x) 0.6666666666666666)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (1.0 / x) * (((cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / cbrt((x * x))) + (0.3333333333333333 * (x * sqrt((x * (1.0 / cbrt(x)))))))) / x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt(x)), pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.06172839506172839 / cbrt(Float64(x * x))) + Float64(0.3333333333333333 * Float64(x * sqrt(Float64(x * Float64(1.0 / cbrt(x)))))))) / x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(x)), (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.06172839506172839 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Sqrt[N[(x * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf 32.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. sqrt-unprod 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. sqr-neg 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt{\color{blue}{\left(-\sqrt[3]{x \cdot x}\right) \cdot \left(-\sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{-\sqrt[3]{x \cdot x}} \cdot \sqrt{-\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(-\sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      6. neg-sub0 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(0 - \sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      7. flip3-- 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt[3]{x \cdot x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   7. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{x \cdot x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. +-lft-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. mul0-lft 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{x \cdot \sqrt[3]{x} + \color{blue}{0}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. +-rgt-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   9. Simplified 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{x \cdot \sqrt[3]{x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 5.1%

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

      4. div-sub 5.1%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 5.1%

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

      1. div-sub 5.1%

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

      2. +-commutative 5.1%

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

      3. associate--l+ 5.1%

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

      4. +-inverses 5.1%

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

      5. metadata-eval 5.1%

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

      6. +-commutative 5.1%

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

      7. fma-define 5.1%

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

      8. +-commutative 5.1%

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

      9. +-commutative 5.1%

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

      10. rem-exp-log 5.1%

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

      11. +-commutative 5.1%

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

      12. log1p-undefine 5.1%

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

      13. rem-exp-log 5.1%

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

      14. +-commutative 5.1%

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

      15. log1p-undefine 5.1%

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

   6. Simplified 5.1%

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

      1. pow1/3 5.1%

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

      2. exp-sum 5.1%

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

      3. log1p-undefine 5.1%

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

      4. add-exp-log 5.1%

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

      5. log1p-undefine 5.1%

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

      6. add-exp-log 5.1%

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

      7. unpow-prod-down 91.3%

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

      8. pow-prod-up 91.3%

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

      9. +-commutative 91.3%

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

      10. metadata-eval 91.3%

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

   8. Applied egg-rr 91.3%

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

   9. Taylor expanded in x around inf 91.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (*
    (/ 1.0 x)
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (/ 0.06172839506172839 (cbrt (* x x)))
       (* 0.3333333333333333 (* x (sqrt (* x (/ 1.0 (cbrt x))))))))
     x))
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (1.0 / x) * (((cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / cbrt((x * x))) + (0.3333333333333333 * (x * sqrt((x * (1.0 / cbrt(x)))))))) / x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(1.0 / x) * Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.06172839506172839 / cbrt(Float64(x * x))) + Float64(0.3333333333333333 * Float64(x * sqrt(Float64(x * Float64(1.0 / cbrt(x)))))))) / x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.06172839506172839 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Sqrt[N[(x * N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf 32.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. rem-square-sqrt 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{\sqrt[3]{x \cdot x}} \cdot \sqrt{\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. sqrt-unprod 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\sqrt{\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x}}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. sqr-neg 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt{\color{blue}{\left(-\sqrt[3]{x \cdot x}\right) \cdot \left(-\sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. sqrt-unprod 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{-\sqrt[3]{x \cdot x}} \cdot \sqrt{-\sqrt[3]{x \cdot x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(-\sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      6. neg-sub0 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(0 - \sqrt[3]{x \cdot x}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      7. flip3-- 0.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{{0}^{3} - {\left(\sqrt[3]{x \cdot x}\right)}^{3}}{0 \cdot 0 + \left(\sqrt[3]{x \cdot x} \cdot \sqrt[3]{x \cdot x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   7. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{x \cdot x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. associate-/l* 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{0 + \left(x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}\right)}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      2. +-lft-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x} + 0 \cdot \sqrt[3]{x \cdot x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      3. mul0-lft 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{x \cdot \sqrt[3]{x} + \color{blue}{0}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

      4. +-rgt-identity 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \left(x \cdot \frac{x}{\color{blue}{x \cdot \sqrt[3]{x}}}\right)\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   9. Simplified 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{x}{x \cdot \sqrt[3]{x}}\right)}\right) \cdot 0.1111111111111111}\right)}{{x}^{2}} \]

   11. Applied egg-rr 98.2%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt{x \cdot \frac{1}{\sqrt[3]{x}}}\right)\right)}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 5.1%

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

      4. div-sub 5.1%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 5.1%

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

      1. div-sub 5.1%

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

      2. +-commutative 5.1%

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

      3. associate--l+ 5.1%

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

      4. +-inverses 5.1%

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

      5. metadata-eval 5.1%

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

      6. +-commutative 5.1%

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

      7. fma-define 5.1%

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

      8. +-commutative 5.1%

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

      9. +-commutative 5.1%

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

      10. rem-exp-log 5.1%

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

      11. +-commutative 5.1%

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

      12. log1p-undefine 5.1%

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

      13. rem-exp-log 5.1%

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

      14. +-commutative 5.1%

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

      15. log1p-undefine 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in x around 0 19.9%

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

Alternative 7: 78.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (/ 0.06172839506172839 (cbrt (* x x)))
       (* 0.3333333333333333 (* x (cbrt x)))))
     x)
    x)
   (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (((cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / cbrt((x * x))) + (0.3333333333333333 * (x * cbrt(x))))) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.06172839506172839 / cbrt(Float64(x * x))) + Float64(0.3333333333333333 * Float64(x * cbrt(x))))) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.06172839506172839 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf 32.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. pow2 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 49.8%

         \[\leadsto \color{blue}{\frac{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)}{x}}{x}} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x}}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 5.1%

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

      4. div-sub 5.1%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 5.1%

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

      1. div-sub 5.1%

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

      2. +-commutative 5.1%

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

      3. associate--l+ 5.1%

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

      4. +-inverses 5.1%

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

      5. metadata-eval 5.1%

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

      6. +-commutative 5.1%

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

      7. fma-define 5.1%

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

      8. +-commutative 5.1%

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

      9. +-commutative 5.1%

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

      10. rem-exp-log 5.1%

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

      11. +-commutative 5.1%

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

      12. log1p-undefine 5.1%

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

      13. rem-exp-log 5.1%

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

      14. +-commutative 5.1%

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

      15. log1p-undefine 5.1%

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

   6. Simplified 5.1%

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

   7. Taylor expanded in x around 0 19.9%

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

Alternative 8: 58.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right) + \left(0 - \frac{-0.06172839506172839}{{x}^{0.6666666666666666}}\right)\right)}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (-
    0.0
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (* 0.3333333333333333 (* x (cbrt x)))
       (- 0.0 (/ (- 0.06172839506172839) (pow x 0.6666666666666666)))))
     (* x (- x))))
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.0 - (((cbrt(x) * -0.1111111111111111) + ((0.3333333333333333 * (x * cbrt(x))) + (0.0 - (-0.06172839506172839 / pow(x, 0.6666666666666666))))) / (x * -x));
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.0 - (((Math.cbrt(x) * -0.1111111111111111) + ((0.3333333333333333 * (x * Math.cbrt(x))) + (0.0 - (-0.06172839506172839 / Math.pow(x, 0.6666666666666666))))) / (x * -x));
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.0 - Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.3333333333333333 * Float64(x * cbrt(x))) + Float64(0.0 - Float64(Float64(-0.06172839506172839) / (x ^ 0.6666666666666666))))) / Float64(x * Float64(-x))));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.0 - N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.0 - N[((-0.06172839506172839) / N[Power[x, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.9%

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

   3. Taylor expanded in x around inf 48.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 48.8%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 48.9%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 73.9%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 73.9%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)\right)}}{{x}^{2}} \]

   7. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]
   8. Step-by-step derivation

      1. pow2 97.8%

         \[\leadsto \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{\color{blue}{{x}^{2}}}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      2. pow1/3 97.8%

         \[\leadsto \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\color{blue}{{\left({x}^{2}\right)}^{0.3333333333333333}}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      3. pow-pow 97.8%

         \[\leadsto \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\color{blue}{{x}^{\left(2 \cdot 0.3333333333333333\right)}}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

      4. metadata-eval 97.8%

         \[\leadsto \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{{x}^{\color{blue}{0.6666666666666666}}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   9. Applied egg-rr 97.8%

      \[\leadsto \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\color{blue}{{x}^{0.6666666666666666}}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x} \]

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 4.7%

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

      4. div-sub 4.7%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 4.7%

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

      1. div-sub 4.7%

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

      2. +-commutative 4.7%

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

      3. associate--l+ 4.7%

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

      4. +-inverses 4.7%

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

      5. metadata-eval 4.7%

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

      6. +-commutative 4.7%

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

      7. fma-define 4.7%

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

      8. +-commutative 4.7%

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

      9. +-commutative 4.7%

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

      10. rem-exp-log 4.7%

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

      11. +-commutative 4.7%

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

      12. log1p-undefine 4.7%

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

      13. rem-exp-log 4.7%

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

      14. +-commutative 4.7%

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

      15. log1p-undefine 4.7%

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

   6. Simplified 4.7%

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

      1. pow1/3 4.7%

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

      2. exp-sum 4.7%

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

      3. log1p-undefine 4.7%

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

      4. add-exp-log 4.7%

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

      5. log1p-undefine 4.7%

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

      6. add-exp-log 4.7%

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

      7. unpow-prod-down 91.6%

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

      8. pow-prod-up 91.6%

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

      9. +-commutative 91.6%

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

      10. metadata-eval 91.6%

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

   8. Applied egg-rr 91.6%

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

   9. Taylor expanded in x around 0 17.7%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0 - \frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right) + \left(0 - \frac{-0.06172839506172839}{{x}^{0.6666666666666666}}\right)\right)}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (+
      (* (cbrt x) -0.1111111111111111)
      (+
       (/ 0.06172839506172839 (cbrt (* x x)))
       (* 0.3333333333333333 (* x (cbrt x)))))
     x)
    x)
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (((cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / cbrt((x * x))) + (0.3333333333333333 * (x * cbrt(x))))) / x) / x;
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (((Math.cbrt(x) * -0.1111111111111111) + ((0.06172839506172839 / Math.cbrt((x * x))) + (0.3333333333333333 * (x * Math.cbrt(x))))) / x) / x;
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(Float64(0.06172839506172839 / cbrt(Float64(x * x))) + Float64(0.3333333333333333 * Float64(x * cbrt(x))))) / x) / x);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(N[(0.06172839506172839 / N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.4%

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

   3. Taylor expanded in x around inf 32.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 32.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 32.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 49.0%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. pow2 49.0%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)}{\color{blue}{x \cdot x}} \]

      2. associate-/r* 49.8%

         \[\leadsto \color{blue}{\frac{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)}{x}}{x}} \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x}}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. rem-cube-cbrt 3.1%

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

      3. rem-cube-cbrt 5.1%

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

      4. div-sub 5.1%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 5.1%

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

      1. div-sub 5.1%

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

      2. +-commutative 5.1%

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

      3. associate--l+ 5.1%

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

      4. +-inverses 5.1%

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

      5. metadata-eval 5.1%

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

      6. +-commutative 5.1%

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

      7. fma-define 5.1%

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

      8. +-commutative 5.1%

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

      9. +-commutative 5.1%

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

      10. rem-exp-log 5.1%

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

      11. +-commutative 5.1%

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

      12. log1p-undefine 5.1%

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

      13. rem-exp-log 5.1%

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

      14. +-commutative 5.1%

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

      15. log1p-undefine 5.1%

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

   6. Simplified 5.1%

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

      1. pow1/3 5.1%

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

      2. exp-sum 5.1%

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

      3. log1p-undefine 5.1%

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

      4. add-exp-log 5.1%

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

      5. log1p-undefine 5.1%

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

      6. add-exp-log 5.1%

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

      7. unpow-prod-down 91.3%

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

      8. pow-prod-up 91.3%

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

      9. +-commutative 91.3%

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

      10. metadata-eval 91.3%

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

   8. Applied egg-rr 91.3%

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

   9. Taylor expanded in x around 0 17.7%

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

Alternative 10: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{+161}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.2e+161)
   (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x)))
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 6.2e+161) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) / x));
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (1.0 + cbrt(x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 6.2e+161) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) / x));
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (1.0 + Math.cbrt(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 6.2e+161)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) / x)));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(1.0 + cbrt(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.2e+161], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(1.0 + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.20000000000000013e161

   1. Initial program 8.8%

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

   3. Taylor expanded in x around inf 47.5%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 47.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

      2. sqrt-unprod 47.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

      3. *-commutative 47.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

      4. *-commutative 47.5%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

      5. swap-sqr 47.4%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. Applied egg-rr 71.7%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 71.7%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)\right)}}{{x}^{2}} \]

   7. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

   8. Taylor expanded in x around inf 92.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. unpow2 92.4%

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

      2. associate-/r* 93.8%

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

   10. Simplified 93.8%

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

   if 6.20000000000000013e161 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. rem-cube-cbrt 3.2%

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

      3. rem-cube-cbrt 4.7%

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

      4. div-sub 4.7%

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

      5. cbrt-unprod 1.9%

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

      6. distribute-rgt-out 1.9%

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

      7. +-commutative 1.9%

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

   4. Applied egg-rr 4.7%

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

      1. div-sub 4.7%

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

      2. +-commutative 4.7%

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

      3. associate--l+ 4.7%

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

      4. +-inverses 4.7%

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

      5. metadata-eval 4.7%

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

      6. +-commutative 4.7%

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

      7. fma-define 4.7%

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

      8. +-commutative 4.7%

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

      9. +-commutative 4.7%

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

      10. rem-exp-log 4.7%

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

      11. +-commutative 4.7%

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

      12. log1p-undefine 4.7%

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

      13. rem-exp-log 4.7%

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

      14. +-commutative 4.7%

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

      15. log1p-undefine 4.7%

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

   6. Simplified 4.7%

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

      1. pow1/3 4.7%

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

      2. exp-sum 4.7%

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

      3. log1p-undefine 4.7%

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

      4. add-exp-log 4.7%

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

      5. log1p-undefine 4.7%

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

      6. add-exp-log 4.7%

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

      7. unpow-prod-down 91.6%

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

      8. pow-prod-up 91.6%

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

      9. +-commutative 91.6%

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

      10. metadata-eval 91.6%

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

   8. Applied egg-rr 91.6%

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

   9. Taylor expanded in x around 0 17.7%

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

Alternative 11: 51.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.3333333333333333 (cbrt (/ (/ 1.0 x) x))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf 25.2%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}{{x}^{2}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 25.2%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}} \cdot \sqrt{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}}\right)}{{x}^{2}} \]

   2. sqrt-unprod 25.2%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}}\right)}{{x}^{2}} \]

   3. *-commutative 25.2%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}\right)}\right)}{{x}^{2}} \]

   4. *-commutative 25.2%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

   5. swap-sqr 25.2%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\color{blue}{\left(\sqrt[3]{{x}^{4}} \cdot \sqrt[3]{{x}^{4}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)}{{x}^{2}} \]

5. Applied egg-rr 38.1%

   \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \color{blue}{\sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}}\right)}{{x}^{2}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 38.1%

      \[\leadsto \frac{\color{blue}{1 \cdot \left(-0.1111111111111111 \cdot \sqrt[3]{x} + \left(0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \sqrt{\left(\left(x \cdot x\right) \cdot \sqrt[3]{x \cdot x}\right) \cdot 0.1111111111111111}\right)\right)}}{{x}^{2}} \]

7. Applied egg-rr 51.6%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + \left(\frac{0.06172839506172839}{\sqrt[3]{x \cdot x}} + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)\right)}{x \cdot x}} \]

8. Taylor expanded in x around inf 51.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
9. Step-by-step derivation

   1. unpow2 51.3%

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

   2. associate-/r* 52.1%

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

10. Simplified 52.1%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{\frac{1}{x}}{x}}} \]
11. Add Preprocessing

Alternative 12: 50.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.3333333333333333 (cbrt (/ 1.0 (* x x)))))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around inf 51.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Step-by-step derivation

   1. unpow2 51.3%

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

5. Simplified 51.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}}} \]
6. Add Preprocessing

Alternative 13: 5.4% 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 6.9%

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

3. Taylor expanded in x around 0 1.8%

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

   1. pow1/3 1.8%

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

5. Applied egg-rr 1.8%

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

   1. sqr-pow 1.8%

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

   2. pow-prod-down 1.4%

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

   3. metadata-eval 1.4%

      \[\leadsto 1 - {\left(x \cdot x\right)}^{\color{blue}{0.16666666666666666}} \]

7. Applied egg-rr 1.4%

   \[\leadsto 1 - \color{blue}{{\left(x \cdot x\right)}^{0.16666666666666666}} \]

8. Taylor expanded in x around -inf 5.4%

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

   1. neg-mul-1 5.4%

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

10. Simplified 5.4%

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

11. Final simplification 5.4%

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

Alternative 14: 1.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in x around 0 1.8%

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

Developer target: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (+ (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0))) (* (cbrt x) (cbrt (+ x 1.0)))) (* (cbrt x) (cbrt x))))

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