2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 99.5%

Time: 17.3s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 5.3%

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

   1. flip3-- 5.3%

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

   2. div-inv 5.3%

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

   3. rem-cube-cbrt 5.5%

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

   4. rem-cube-cbrt 7.5%

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

   5. +-commutative 7.5%

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

   6. distribute-rgt-out 7.5%

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

   7. +-commutative 7.5%

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

   8. fma-define 7.5%

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

   9. add-exp-log 7.4%

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

4. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 7.4%

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

   2. *-rgt-identity 7.4%

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

   3. +-commutative 7.4%

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

   4. associate--l+ 93.1%

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

   5. +-inverses 93.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. metadata-eval 93.1%

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

   7. +-commutative 93.1%

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

   8. exp-prod 92.3%

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

6. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 92.3%

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

   2. unpow-prod-down 94.1%

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

8. Applied egg-rr 94.1%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)} \]
9. Step-by-step derivation

   1. pow-sqr 94.1%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]

10. Simplified 94.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
11. Step-by-step derivation

    1. sqr-pow 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}}\right)} \]

    2. pow2 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}\right)}^{2}}\right)} \]

    3. pow-to-exp 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}}\right)}}^{2}\right)} \]

    4. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 2}}{2}}\right)}^{2}\right)} \]

    5. associate-/l* 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{2}{2}\right)}}\right)}^{2}\right)} \]

    6. metadata-eval 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \color{blue}{1}\right)}\right)}^{2}\right)} \]

    7. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(1 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)}^{2}\right)} \]

    8. *-un-lft-identity 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right)}^{2}\right)} \]

    9. pow1/2 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    10. log-pow 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    11. rem-log-exp 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\left(0.5 \cdot \color{blue}{0.6666666666666666}\right) \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    12. metadata-eval 93.1%

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

    13. log1p-undefine 93.1%

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

    14. log-pow 93.5%

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

    15. pow1/3 94.1%

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

    16. add-exp-log 98.4%

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

    17. pow2 98.4%

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

12. Applied egg-rr 98.4%

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

    1. flip3-+ 98.4%

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

    2. +-commutative 98.4%

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

    3. rem-cube-cbrt 99.0%

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

    4. rem-cube-cbrt 99.4%

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

    5. +-commutative 99.4%

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

    6. +-commutative 99.4%

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

    7. +-commutative 99.4%

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

    8. pow2 99.4%

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

    9. +-commutative 99.4%

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

    10. distribute-rgt-out-- 99.4%

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

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

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 / (x ^ 2.0)))))))
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[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.3%

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

   1. flip3-- 5.3%

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

   2. div-inv 5.3%

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

   3. rem-cube-cbrt 5.5%

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

   4. rem-cube-cbrt 7.5%

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

   5. +-commutative 7.5%

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

   6. distribute-rgt-out 7.5%

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

   7. +-commutative 7.5%

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

   8. fma-define 7.5%

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

   9. add-exp-log 7.4%

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

4. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 7.4%

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

   2. *-rgt-identity 7.4%

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

   3. +-commutative 7.4%

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

   4. associate--l+ 93.1%

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

   5. +-inverses 93.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. metadata-eval 93.1%

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

   7. +-commutative 93.1%

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

   8. exp-prod 92.3%

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

6. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 92.3%

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

   2. unpow-prod-down 94.1%

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

8. Applied egg-rr 94.1%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)} \]
9. Step-by-step derivation

   1. pow-sqr 94.1%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]

10. Simplified 94.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
11. Step-by-step derivation

    1. sqr-pow 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}}\right)} \]

    2. pow2 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}\right)}^{2}}\right)} \]

    3. pow-to-exp 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}}\right)}}^{2}\right)} \]

    4. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 2}}{2}}\right)}^{2}\right)} \]

    5. associate-/l* 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{2}{2}\right)}}\right)}^{2}\right)} \]

    6. metadata-eval 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \color{blue}{1}\right)}\right)}^{2}\right)} \]

    7. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(1 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)}^{2}\right)} \]

    8. *-un-lft-identity 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right)}^{2}\right)} \]

    9. pow1/2 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    10. log-pow 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    11. rem-log-exp 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\left(0.5 \cdot \color{blue}{0.6666666666666666}\right) \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    12. metadata-eval 93.1%

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

    13. log1p-undefine 93.1%

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

    14. log-pow 93.5%

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

    15. pow1/3 94.1%

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

    16. add-exp-log 98.4%

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

    17. pow2 98.4%

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

12. Applied egg-rr 98.4%

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

13. Taylor expanded in x around inf 98.6%

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

    1. associate-*r/ 98.6%

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

    2. metadata-eval 98.6%

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

15. Simplified 98.6%

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

16. Final simplification 98.6%

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

Alternative 3: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), Float64(t_0 * t_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[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 5.3%

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

   1. flip3-- 5.3%

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

   2. div-inv 5.3%

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

   3. rem-cube-cbrt 5.5%

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

   4. rem-cube-cbrt 7.5%

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

   5. +-commutative 7.5%

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

   6. distribute-rgt-out 7.5%

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

   7. +-commutative 7.5%

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

   8. fma-define 7.5%

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

   9. add-exp-log 7.4%

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

4. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 7.4%

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

   2. *-rgt-identity 7.4%

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

   3. +-commutative 7.4%

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

   4. associate--l+ 93.1%

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

   5. +-inverses 93.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. metadata-eval 93.1%

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

   7. +-commutative 93.1%

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

   8. exp-prod 92.3%

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

6. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 92.3%

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

   2. unpow-prod-down 94.1%

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

8. Applied egg-rr 94.1%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)} \]
9. Step-by-step derivation

   1. pow-sqr 94.1%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]

10. Simplified 94.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
11. Step-by-step derivation

    1. sqr-pow 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}}\right)} \]

    2. pow2 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}\right)}^{2}}\right)} \]

    3. pow-to-exp 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}}\right)}}^{2}\right)} \]

    4. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 2}}{2}}\right)}^{2}\right)} \]

    5. associate-/l* 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{2}{2}\right)}}\right)}^{2}\right)} \]

    6. metadata-eval 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \color{blue}{1}\right)}\right)}^{2}\right)} \]

    7. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(1 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)}^{2}\right)} \]

    8. *-un-lft-identity 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right)}^{2}\right)} \]

    9. pow1/2 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    10. log-pow 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    11. rem-log-exp 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\left(0.5 \cdot \color{blue}{0.6666666666666666}\right) \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    12. metadata-eval 93.1%

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

    13. log1p-undefine 93.1%

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

    14. log-pow 93.5%

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

    15. pow1/3 94.1%

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

    16. add-exp-log 98.4%

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

    17. pow2 98.4%

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

12. Applied egg-rr 98.4%

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

13. Final simplification 98.4%

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

Alternative 4: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (cbrt (* x -0.0013717421124828531)))
     x)
    x)
   (/
    1.0
    (+
     (exp (* (log1p x) 0.6666666666666666))
     (* (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), cbrt((x * -0.0013717421124828531))) / x) / x;
	} else {
		tmp = 1.0 / (exp((log1p(x) * 0.6666666666666666)) + (cbrt(x) * (cbrt(x) + cbrt((1.0 + x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), cbrt(Float64(x * -0.0013717421124828531))) / x) / x);
	else
		tmp = Float64(1.0 / Float64(exp(Float64(log1p(x) * 0.6666666666666666)) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(Float64(1.0 + x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[(x * -0.0013717421124828531), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[Exp[N[(N[Log[1 + x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf 32.4%

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

      1. +-commutative 32.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}} + -0.1111111111111111 \cdot \sqrt[3]{x}}}{{x}^{2}} \]

      2. fma-define 32.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. Simplified 32.4%

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

      1. metadata-eval 32.4%

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

      2. pow-prod-up 32.4%

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

      3. cbrt-prod 65.6%

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

      4. pow2 65.6%

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

      5. unpow2 65.6%

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

      6. cbrt-prod 66.4%

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

      7. pow2 66.4%

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

   7. Applied egg-rr 66.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{2}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. fma-undefine 66.4%

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

      2. +-commutative 66.4%

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

      3. *-commutative 66.4%

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

      4. pow-pow 66.4%

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

      5. metadata-eval 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. *-un-lft-identity 66.4%

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

       2. unpow2 66.4%

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

       3. times-frac 97.4%

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

       4. +-commutative 97.4%

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

       5. fma-define 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{x} \]

       6. add-cbrt-cube 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{x} \cdot -0.1111111111111111\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}}\right)}{x} \]

       7. pow3 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}^{3}}}\right)}{x} \]

       8. unpow-prod-down 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x}\right)}^{3} \cdot {-0.1111111111111111}^{3}}}\right)}{x} \]

       9. rem-cube-cbrt 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{x} \cdot {-0.1111111111111111}^{3}}\right)}{x} \]

       10. metadata-eval 97.4%

           \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot \color{blue}{-0.0013717421124828531}}\right)}{x} \]

   11. Applied egg-rr 97.4%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*l/ 97.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

       2. *-lft-identity 97.4%

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}}{x} \]

   13. Simplified 97.4%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.2%

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

      1. flip3-- 5.2%

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

      2. div-inv 5.2%

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

      3. rem-cube-cbrt 3.3%

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

      4. rem-cube-cbrt 5.2%

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

      5. +-commutative 5.2%

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

      6. distribute-rgt-out 5.2%

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

      7. +-commutative 5.2%

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

      8. fma-define 5.2%

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

      9. add-exp-log 5.2%

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

   4. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. +-commutative 5.2%

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

      4. associate--l+ 91.3%

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

      5. +-inverses 91.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. exp-prod 90.5%

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

   6. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. fma-undefine 90.5%

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

      2. +-commutative 90.5%

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

      3. +-commutative 90.5%

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

      4. +-commutative 90.5%

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

   8. Applied egg-rr 90.5%

      \[\leadsto \frac{1}{\color{blue}{{\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
   9. Step-by-step derivation

      1. add-exp-log 90.3%

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

      2. log-pow 91.3%

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

      3. rem-log-exp 91.3%

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

   10. Applied egg-rr 91.3%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (cbrt (* x -0.0013717421124828531)))
     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 = (fma(0.3333333333333333, pow(cbrt(x), 4.0), cbrt((x * -0.0013717421124828531))) / 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(fma(0.3333333333333333, (cbrt(x) ^ 4.0), cbrt(Float64(x * -0.0013717421124828531))) / 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[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[(x * -0.0013717421124828531), $MachinePrecision], 1/3], $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] + 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 5.4%

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

   3. Taylor expanded in x around inf 32.4%

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

      1. +-commutative 32.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}} + -0.1111111111111111 \cdot \sqrt[3]{x}}}{{x}^{2}} \]

      2. fma-define 32.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. Simplified 32.4%

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

      1. metadata-eval 32.4%

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

      2. pow-prod-up 32.4%

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

      3. cbrt-prod 65.6%

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

      4. pow2 65.6%

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

      5. unpow2 65.6%

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

      6. cbrt-prod 66.4%

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

      7. pow2 66.4%

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

   7. Applied egg-rr 66.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{2}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. fma-undefine 66.4%

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

      2. +-commutative 66.4%

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

      3. *-commutative 66.4%

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

      4. pow-pow 66.4%

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

      5. metadata-eval 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. *-un-lft-identity 66.4%

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

       2. unpow2 66.4%

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

       3. times-frac 97.4%

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

       4. +-commutative 97.4%

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

       5. fma-define 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{x} \]

       6. add-cbrt-cube 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{x} \cdot -0.1111111111111111\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}}\right)}{x} \]

       7. pow3 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}^{3}}}\right)}{x} \]

       8. unpow-prod-down 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x}\right)}^{3} \cdot {-0.1111111111111111}^{3}}}\right)}{x} \]

       9. rem-cube-cbrt 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{x} \cdot {-0.1111111111111111}^{3}}\right)}{x} \]

       10. metadata-eval 97.4%

           \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot \color{blue}{-0.0013717421124828531}}\right)}{x} \]

   11. Applied egg-rr 97.4%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*l/ 97.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

       2. *-lft-identity 97.4%

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}}{x} \]

   13. Simplified 97.4%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.2%

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

      1. flip3-- 5.2%

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

      2. div-inv 5.2%

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

      3. rem-cube-cbrt 3.3%

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

      4. rem-cube-cbrt 5.2%

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

      5. +-commutative 5.2%

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

      6. distribute-rgt-out 5.2%

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

      7. +-commutative 5.2%

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

      8. fma-define 5.2%

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

      9. add-exp-log 5.2%

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

   4. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. +-commutative 5.2%

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

      4. associate--l+ 91.3%

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

      5. +-inverses 91.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. exp-prod 90.5%

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

   6. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 90.5%

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

      2. unpow-prod-down 91.6%

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

   8. Applied egg-rr 91.6%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)} \]
   9. Step-by-step derivation

      1. pow-sqr 91.6%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]

   10. Simplified 91.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
   11. Step-by-step derivation

       1. sqr-pow 91.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}}\right)} \]

       2. pow2 91.6%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}\right)}^{2}}\right)} \]

       3. pow-to-exp 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}}\right)}}^{2}\right)} \]

       4. *-commutative 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 2}}{2}}\right)}^{2}\right)} \]

       5. associate-/l* 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{2}{2}\right)}}\right)}^{2}\right)} \]

       6. metadata-eval 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \color{blue}{1}\right)}\right)}^{2}\right)} \]

       7. *-commutative 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(1 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)}^{2}\right)} \]

       8. *-un-lft-identity 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right)}^{2}\right)} \]

       9. pow1/2 91.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

       10. log-pow 91.3%

           \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

       11. rem-log-exp 91.3%

           \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\left(0.5 \cdot \color{blue}{0.6666666666666666}\right) \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

       12. metadata-eval 91.3%

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

       13. log1p-undefine 91.3%

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

       14. log-pow 91.8%

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

       15. pow1/3 92.8%

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

       16. add-exp-log 98.4%

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

       17. pow2 98.4%

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

   12. Applied egg-rr 98.4%

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

       1. pow2 98.4%

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

       2. pow1/3 91.0%

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

       3. pow-pow 91.0%

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

       4. metadata-eval 91.0%

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

   14. Applied egg-rr 91.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 5.3%

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

   1. flip3-- 5.3%

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

   2. div-inv 5.3%

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

   3. rem-cube-cbrt 5.5%

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

   4. rem-cube-cbrt 7.5%

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

   5. +-commutative 7.5%

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

   6. distribute-rgt-out 7.5%

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

   7. +-commutative 7.5%

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

   8. fma-define 7.5%

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

   9. add-exp-log 7.4%

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

4. Applied egg-rr 7.4%

   \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 7.4%

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

   2. *-rgt-identity 7.4%

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

   3. +-commutative 7.4%

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

   4. associate--l+ 93.1%

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

   5. +-inverses 93.1%

      \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

   6. metadata-eval 93.1%

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

   7. +-commutative 93.1%

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

   8. exp-prod 92.3%

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

6. Simplified 92.3%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 92.3%

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

   2. unpow-prod-down 94.1%

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

8. Applied egg-rr 94.1%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}}\right)} \]
9. Step-by-step derivation

   1. pow-sqr 94.1%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]

10. Simplified 94.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(2 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)} \]
11. Step-by-step derivation

    1. sqr-pow 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)} \cdot {\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}}\right)} \]

    2. pow2 94.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left({\left(\sqrt{e^{0.6666666666666666}}\right)}^{\left(\frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}\right)}\right)}^{2}}\right)} \]

    3. pow-to-exp 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\color{blue}{\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{2 \cdot \mathsf{log1p}\left(x\right)}{2}}\right)}}^{2}\right)} \]

    4. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \frac{\color{blue}{\mathsf{log1p}\left(x\right) \cdot 2}}{2}}\right)}^{2}\right)} \]

    5. associate-/l* 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \frac{2}{2}\right)}}\right)}^{2}\right)} \]

    6. metadata-eval 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \color{blue}{1}\right)}\right)}^{2}\right)} \]

    7. *-commutative 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\left(1 \cdot \mathsf{log1p}\left(x\right)\right)}}\right)}^{2}\right)} \]

    8. *-un-lft-identity 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(\sqrt{e^{0.6666666666666666}}\right) \cdot \color{blue}{\mathsf{log1p}\left(x\right)}}\right)}^{2}\right)} \]

    9. pow1/2 93.1%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    10. log-pow 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)} \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    11. rem-log-exp 93.1%

        \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\left(0.5 \cdot \color{blue}{0.6666666666666666}\right) \cdot \mathsf{log1p}\left(x\right)}\right)}^{2}\right)} \]

    12. metadata-eval 93.1%

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

    13. log1p-undefine 93.1%

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

    14. log-pow 93.5%

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

    15. pow1/3 94.1%

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

    16. add-exp-log 98.4%

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

    17. pow2 98.4%

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

12. Applied egg-rr 98.4%

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

13. Taylor expanded in x around inf 97.9%

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

14. Final simplification 97.9%

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

Alternative 7: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt{0.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (cbrt (* x -0.0013717421124828531)))
     x)
    x)
   (pow (* (cbrt (/ 1.0 x)) (sqrt 0.5)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), cbrt((x * -0.0013717421124828531))) / x) / x;
	} else {
		tmp = pow((cbrt((1.0 / x)) * sqrt(0.5)), 2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), cbrt(Float64(x * -0.0013717421124828531))) / x) / x);
	else
		tmp = Float64(cbrt(Float64(1.0 / x)) * sqrt(0.5)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[Power[N[(x * -0.0013717421124828531), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[Power[N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.4%

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

   3. Taylor expanded in x around inf 32.4%

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

      1. +-commutative 32.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}} + -0.1111111111111111 \cdot \sqrt[3]{x}}}{{x}^{2}} \]

      2. fma-define 32.4%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, \sqrt[3]{{x}^{4}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}}{{x}^{2}} \]

   5. Simplified 32.4%

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

      1. metadata-eval 32.4%

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

      2. pow-prod-up 32.4%

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

      3. cbrt-prod 65.6%

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

      4. pow2 65.6%

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

      5. unpow2 65.6%

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

      6. cbrt-prod 66.4%

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

      7. pow2 66.4%

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

   7. Applied egg-rr 66.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{{\left({\left(\sqrt[3]{x}\right)}^{2}\right)}^{2}}, -0.1111111111111111 \cdot \sqrt[3]{x}\right)}{{x}^{2}} \]
   8. Step-by-step derivation

      1. fma-undefine 66.4%

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

      2. +-commutative 66.4%

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

      3. *-commutative 66.4%

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

      4. pow-pow 66.4%

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

      5. metadata-eval 66.4%

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

   9. Applied egg-rr 66.4%

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

       1. *-un-lft-identity 66.4%

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

       2. unpow2 66.4%

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

       3. times-frac 97.4%

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

       4. +-commutative 97.4%

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

       5. fma-define 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}{x} \]

       6. add-cbrt-cube 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \color{blue}{\sqrt[3]{\left(\left(\sqrt[3]{x} \cdot -0.1111111111111111\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)\right) \cdot \left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}}\right)}{x} \]

       7. pow3 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x} \cdot -0.1111111111111111\right)}^{3}}}\right)}{x} \]

       8. unpow-prod-down 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{{\left(\sqrt[3]{x}\right)}^{3} \cdot {-0.1111111111111111}^{3}}}\right)}{x} \]

       9. rem-cube-cbrt 97.4%

          \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{\color{blue}{x} \cdot {-0.1111111111111111}^{3}}\right)}{x} \]

       10. metadata-eval 97.4%

           \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot \color{blue}{-0.0013717421124828531}}\right)}{x} \]

   11. Applied egg-rr 97.4%

       \[\leadsto \color{blue}{\frac{1}{x} \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}} \]
   12. Step-by-step derivation

       1. associate-*l/ 97.4%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

       2. *-lft-identity 97.4%

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}}{x} \]

   13. Simplified 97.4%

       \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x \cdot -0.0013717421124828531}\right)}{x}}{x}} \]

   if 1.54999999999999995e231 < x

   1. Initial program 5.2%

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

      1. flip3-- 5.2%

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

      2. div-inv 5.2%

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

      3. rem-cube-cbrt 3.3%

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

      4. rem-cube-cbrt 5.2%

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

      5. +-commutative 5.2%

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

      6. distribute-rgt-out 5.2%

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

      7. +-commutative 5.2%

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

      8. fma-define 5.2%

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

      9. add-exp-log 5.2%

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

   4. Applied egg-rr 5.2%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. +-commutative 5.2%

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

      4. associate--l+ 91.3%

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

      5. +-inverses 91.3%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.3%

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

      7. +-commutative 91.3%

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

      8. exp-prod 90.5%

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

   6. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 90.5%

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

      2. pow2 90.5%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}}\right)}^{2}} \]

      3. inv-pow 90.5%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-1}}}\right)}^{2} \]

      4. sqrt-pow1 90.5%

         \[\leadsto {\color{blue}{\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

      5. +-commutative 90.5%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x} + \sqrt[3]{1 + x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2} \]

      6. +-commutative 90.5%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{\color{blue}{x + 1}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2} \]

      7. metadata-eval 90.5%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   8. Applied egg-rr 90.5%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-0.5}\right)}^{2}} \]

   9. Taylor expanded in x around inf 20.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt{0.5}\right)}}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt{0.5}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0))))
   (pow (* (cbrt (/ 1.0 x)) (sqrt 0.5)) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / pow(x, 2.0)));
	} else {
		tmp = pow((cbrt((1.0 / x)) * sqrt(0.5)), 2.0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * Math.cbrt((1.0 / Math.pow(x, 2.0)));
	} else {
		tmp = Math.pow((Math.cbrt((1.0 / x)) * Math.sqrt(0.5)), 2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0))));
	else
		tmp = Float64(cbrt(Float64(1.0 / x)) * sqrt(0.5)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 5.9%

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

   3. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.7%

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

      3. rem-cube-cbrt 3.3%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-inverses 91.7%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.7%

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

      7. +-commutative 91.7%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 90.9%

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

      2. pow2 90.9%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}}\right)}^{2}} \]

      3. inv-pow 90.9%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-1}}}\right)}^{2} \]

      4. sqrt-pow1 90.9%

         \[\leadsto {\color{blue}{\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{2} \]

      5. +-commutative 90.9%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x} + \sqrt[3]{1 + x}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2} \]

      6. +-commutative 90.9%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{\color{blue}{x + 1}}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\left(\frac{-1}{2}\right)}\right)}^{2} \]

      7. metadata-eval 90.9%

         \[\leadsto {\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   8. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, {\left(e^{0.6666666666666666}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)\right)}^{-0.5}\right)}^{2}} \]

   9. Taylor expanded in x around inf 20.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\frac{1}{x}} \cdot \sqrt{0.5}\right)}}^{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (* 0.3333333333333333 (cbrt (/ 1.0 (pow x 2.0))))
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ 1.0 (cbrt x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / pow(x, 2.0)));
	} 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 = 0.3333333333333333 * Math.cbrt((1.0 / Math.pow(x, 2.0)));
	} 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(0.3333333333333333 * cbrt(Float64(1.0 / (x ^ 2.0))));
	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[(0.3333333333333333 * N[Power[N[(1.0 / N[Power[x, 2.0], $MachinePrecision]), $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 < 1.35000000000000003e154

   1. Initial program 5.9%

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

   3. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]

   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. div-inv 4.7%

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

      3. rem-cube-cbrt 3.3%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) - x\right) \cdot \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-inverses 91.7%

         \[\leadsto \frac{1 + \color{blue}{0}}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)} \]

      6. metadata-eval 91.7%

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

      7. +-commutative 91.7%

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

      8. exp-prod 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in x around 0 17.7%

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

Alternative 10: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 52.1%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}} \]
4. Add Preprocessing

Alternative 11: 5.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{x} - {x}^{0.3333333333333333} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 4.1%

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

   1. pow1/3 5.8%

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

5. Applied egg-rr 5.8%

   \[\leadsto \sqrt[3]{x} - \color{blue}{{x}^{0.3333333333333333}} \]
6. Add Preprocessing

Alternative 12: 5.3% 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 5.3%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.2%

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

   5. fabs-neg 5.2%

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

   6. unpow1/3 5.2%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.3333333333333333}}\right| \]

   7. metadata-eval 5.2%

      \[\leadsto 1 + \left|{x}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}}\right| \]

   8. pow-sqr 5.2%

      \[\leadsto 1 + \left|\color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}}\right| \]

   9. fabs-sqr 5.2%

      \[\leadsto 1 + \color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}} \]

   10. pow-sqr 5.2%

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

   11. metadata-eval 5.2%

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

   12. unpow1/3 5.2%

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

5. Simplified 5.2%

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

Alternative 13: 4.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 5.3%

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

3. Taylor expanded in x around inf 4.1%

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

4. Taylor expanded in x around 0 4.1%

   \[\leadsto \color{blue}{0} \]
5. 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 2024110 
(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)))