2cbrt (problem 3.3.4)

Percentage Accurate: 6.7% → 96.0%

Time: 20.2s

Alternatives: 15

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: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{4}\\ \mathbf{if}\;x \leq 1.45 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{t\_0 \cdot \frac{{\left(\sqrt[3]{x}\right)}^{2}}{\mathsf{fma}\left(0.3333333333333333, t\_0, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 4.0)))
   (if (<= x 1.45e+231)
     (/
      1.0
      (*
       t_0
       (/
        (pow (cbrt x) 2.0)
        (fma 0.3333333333333333 t_0 (* (cbrt x) -0.1111111111111111)))))
     (/
      1.0
      (fma
       (cbrt x)
       (+ (cbrt x) (cbrt (+ x 1.0)))
       (/ 1.0 (pow (exp -0.6666666666666666) (log x))))))))

C

double code(double x) {
	double t_0 = pow(cbrt(x), 4.0);
	double tmp;
	if (x <= 1.45e+231) {
		tmp = 1.0 / (t_0 * (pow(cbrt(x), 2.0) / fma(0.3333333333333333, t_0, (cbrt(x) * -0.1111111111111111))));
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), (1.0 / pow(exp(-0.6666666666666666), log(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(x) ^ 4.0
	tmp = 0.0
	if (x <= 1.45e+231)
		tmp = Float64(1.0 / Float64(t_0 * Float64((cbrt(x) ^ 2.0) / fma(0.3333333333333333, t_0, Float64(cbrt(x) * -0.1111111111111111)))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), Float64(1.0 / (exp(-0.6666666666666666) ^ log(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[x, 1.45e+231], N[(1.0 / N[(t$95$0 * N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(0.3333333333333333 * t$95$0 + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[Exp[-0.6666666666666666], $MachinePrecision], N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 98.2%

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

   if 1.45e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+231}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{x}\right)}^{4} \cdot \frac{{\left(\sqrt[3]{x}\right)}^{2}}{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{4}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, t\_0, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{t\_0}}{\sqrt[3]{x}}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 4.0)))
   (if (<= x 1.5e+231)
     (/
      (/
       (/ (fma 0.3333333333333333 t_0 (* (cbrt x) -0.1111111111111111)) t_0)
       (cbrt x))
      (cbrt x))
     (/
      1.0
      (fma
       (cbrt x)
       (+ (cbrt x) (cbrt (+ x 1.0)))
       (/ 1.0 (pow (exp -0.6666666666666666) (log x))))))))

C

double code(double x) {
	double t_0 = pow(cbrt(x), 4.0);
	double tmp;
	if (x <= 1.5e+231) {
		tmp = ((fma(0.3333333333333333, t_0, (cbrt(x) * -0.1111111111111111)) / t_0) / cbrt(x)) / cbrt(x);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), (1.0 / pow(exp(-0.6666666666666666), log(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(x) ^ 4.0
	tmp = 0.0
	if (x <= 1.5e+231)
		tmp = Float64(Float64(Float64(fma(0.3333333333333333, t_0, Float64(cbrt(x) * -0.1111111111111111)) / t_0) / cbrt(x)) / cbrt(x));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), Float64(1.0 / (exp(-0.6666666666666666) ^ log(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[x, 1.5e+231], N[(N[(N[(N[(0.3333333333333333 * t$95$0 + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[Exp[-0.6666666666666666], $MachinePrecision], N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 98.2%

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

   if 1.5000000000000001e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{{\left(\sqrt[3]{x}\right)}^{4}}}{\sqrt[3]{x}}}{\sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{4}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, t\_0, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{t\_0} \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 4.0)))
   (if (<= x 1.5e+231)
     (*
      (/ (fma 0.3333333333333333 t_0 (* (cbrt x) -0.1111111111111111)) t_0)
      (pow (cbrt x) -2.0))
     (/
      1.0
      (fma
       (cbrt x)
       (+ (cbrt x) (cbrt (+ x 1.0)))
       (/ 1.0 (pow (exp -0.6666666666666666) (log x))))))))

C

double code(double x) {
	double t_0 = pow(cbrt(x), 4.0);
	double tmp;
	if (x <= 1.5e+231) {
		tmp = (fma(0.3333333333333333, t_0, (cbrt(x) * -0.1111111111111111)) / t_0) * pow(cbrt(x), -2.0);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), (1.0 / pow(exp(-0.6666666666666666), log(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(x) ^ 4.0
	tmp = 0.0
	if (x <= 1.5e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, t_0, Float64(cbrt(x) * -0.1111111111111111)) / t_0) * (cbrt(x) ^ -2.0));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), Float64(1.0 / (exp(-0.6666666666666666) ^ log(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[x, 1.5e+231], N[(N[(N[(0.3333333333333333 * t$95$0 + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[Power[N[Power[x, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[Exp[-0.6666666666666666], $MachinePrecision], N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. pow1/3 30.5%

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

      2. pow-pow 64.8%

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

      3. metadata-eval 64.8%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}}{{x}^{2}} \]

   5. Applied egg-rr 64.8%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{1.3333333333333333}}}{{x}^{2}} \]

   7. Applied egg-rr 98.2%

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

   if 1.5000000000000001e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{{\left(\sqrt[3]{x}\right)}^{4}} \cdot {\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt[3]{x}\right)}^{4}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, t\_0, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{t\_0}}{{\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (cbrt x) 4.0)))
   (if (<= x 1.5e+231)
     (/
      (/ (fma 0.3333333333333333 t_0 (* (cbrt x) -0.1111111111111111)) t_0)
      (pow (cbrt x) 2.0))
     (/
      1.0
      (fma
       (cbrt x)
       (+ (cbrt x) (cbrt (+ x 1.0)))
       (/ 1.0 (pow (exp -0.6666666666666666) (log x))))))))

C

double code(double x) {
	double t_0 = pow(cbrt(x), 4.0);
	double tmp;
	if (x <= 1.5e+231) {
		tmp = (fma(0.3333333333333333, t_0, (cbrt(x) * -0.1111111111111111)) / t_0) / pow(cbrt(x), 2.0);
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), (1.0 / pow(exp(-0.6666666666666666), log(x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(x) ^ 4.0
	tmp = 0.0
	if (x <= 1.5e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, t_0, Float64(cbrt(x) * -0.1111111111111111)) / t_0) / (cbrt(x) ^ 2.0));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), Float64(1.0 / (exp(-0.6666666666666666) ^ log(x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[x, 1.5e+231], N[(N[(N[(0.3333333333333333 * t$95$0 + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[Exp[-0.6666666666666666], $MachinePrecision], N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 98.2%

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

   if 1.5000000000000001e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{{\left(\sqrt[3]{x}\right)}^{4}}}{{\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (* (cbrt x) -0.1111111111111111))
     x)
    x)
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (/ 1.0 (pow (exp -0.6666666666666666) (log x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), (cbrt(x) * -0.1111111111111111)) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), (1.0 / pow(exp(-0.6666666666666666), log(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), Float64(cbrt(x) * -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), Float64(1.0 / (exp(-0.6666666666666666) ^ log(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5e+231], N[(N[(N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $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[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[Exp[-0.6666666666666666], $MachinePrecision], N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 97.5%

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

   if 1.5000000000000001e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, \frac{1}{{\left(e^{-0.6666666666666666}\right)}^{\log x}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (* (cbrt x) -0.1111111111111111))
     x)
    x)
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (exp (* 0.6666666666666666 (log1p x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), (cbrt(x) * -0.1111111111111111)) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), exp((0.6666666666666666 * log1p(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), Float64(cbrt(x) * -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), exp(Float64(0.6666666666666666 * log1p(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[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $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[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 97.5%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. pow-exp 91.3%

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

   8. Applied egg-rr 91.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{\log x \cdot 0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (* (cbrt x) -0.1111111111111111))
     x)
    x)
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ x 1.0)))
     (exp (* (log 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) * -0.1111111111111111)) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((x + 1.0))), exp((log(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), Float64(cbrt(x) * -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), exp(Float64(log(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[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $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[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(N[Log[x], $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 97.5%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

      1. add-sqr-sqrt 90.4%

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

      2. unpow-prod-down 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. pow-sqr 92.2%

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

   10. Simplified 92.2%

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

   11. Taylor expanded in x around inf 91.3%

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

       1. exp-prod 90.4%

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

       2. log-rec 90.4%

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

   13. Simplified 90.4%

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

       1. pow-unpow 92.2%

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

       2. pow-neg 92.3%

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

       3. pow-exp 92.3%

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

       4. pow1/2 92.3%

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

       5. log-pow 92.3%

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

       6. rem-log-exp 92.3%

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

       7. metadata-eval 92.3%

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

       8. metadata-eval 92.3%

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

   15. Applied egg-rr 92.3%

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

       1. exp-prod 91.3%

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

       2. rec-exp 91.3%

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

       3. rem-log-exp 91.3%

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

       4. *-commutative 91.3%

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

       5. distribute-rgt-neg-in 91.3%

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

       6. rem-log-exp 91.3%

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

       7. metadata-eval 91.3%

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

   17. Simplified 91.3%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x + 1}, e^{\log x \cdot 0.6666666666666666}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right), \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (* (cbrt x) -0.1111111111111111))
     x)
    x)
   (/ 1.0 (fma (expm1 (log1p (cbrt x))) (+ (cbrt x) (cbrt (+ x 1.0))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), (cbrt(x) * -0.1111111111111111)) / x) / x;
	} else {
		tmp = 1.0 / fma(expm1(log1p(cbrt(x))), (cbrt(x) + cbrt((x + 1.0))), 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), Float64(cbrt(x) * -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(1.0 / fma(expm1(log1p(cbrt(x))), Float64(cbrt(x) + cbrt(Float64(x + 1.0))), 1.0));
	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[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[(Exp[N[Log[1 + N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 97.5%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in x around 0 20.0%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}, \sqrt[3]{1 + x} + \sqrt[3]{x}, 1\right)} \]

      2. expm1-undefine 20.0%

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

   9. Applied egg-rr 20.0%

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

       1. expm1-define 20.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}, \sqrt[3]{1 + x} + \sqrt[3]{x}, 1\right)} \]

   11. Simplified 20.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right)}, \sqrt[3]{1 + x} + \sqrt[3]{x}, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{x}\right)\right), \sqrt[3]{x} + \sqrt[3]{x + 1}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      0.3333333333333333
      (pow (cbrt x) 4.0)
      (* (cbrt x) -0.1111111111111111))
     x)
    x)
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ (cbrt x) (cbrt (+ x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(0.3333333333333333, pow(cbrt(x), 4.0), (cbrt(x) * -0.1111111111111111)) / x) / x;
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (cbrt(x) + cbrt((x + 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(0.3333333333333333, (cbrt(x) ^ 4.0), Float64(cbrt(x) * -0.1111111111111111)) / x) / x);
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(Float64(x + 1.0))))));
	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[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 5.8%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. *-un-lft-identity 32.8%

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

      2. add-cbrt-cube 22.1%

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

      3. pow-sqr 22.2%

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

      4. metadata-eval 22.2%

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

      5. cbrt-prod 31.8%

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

      6. unpow2 31.8%

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

      7. cbrt-prod 31.6%

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

      8. times-frac 31.5%

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

   5. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{{x}^{4}}} \cdot \frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, \sqrt[3]{{x}^{4} \cdot 0.037037037037037035}\right)}{{\left(\sqrt[3]{x}\right)}^{2}}} \]

   7. Applied egg-rr 97.5%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 2.9%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      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 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in x around 0 20.0%

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

      1. fma-undefine 20.0%

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

      2. +-commutative 20.0%

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

      3. +-commutative 20.0%

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

   9. Applied egg-rr 20.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(0.3333333333333333, {\left(\sqrt[3]{x}\right)}^{4}, \sqrt[3]{x} \cdot -0.1111111111111111\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/
    (+ (* (cbrt x) -0.1111111111111111) (* 0.3333333333333333 (* x (cbrt x))))
    (pow x 2.0))
   (/ 1.0 (+ 1.0 (* (cbrt x) (+ (cbrt x) (cbrt (+ x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = ((cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * (x * cbrt(x)))) / pow(x, 2.0);
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (cbrt(x) + cbrt((x + 1.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = ((Math.cbrt(x) * -0.1111111111111111) + (0.3333333333333333 * (x * Math.cbrt(x)))) / Math.pow(x, 2.0);
	} else {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (Math.cbrt(x) + Math.cbrt((x + 1.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(Float64(cbrt(x) * -0.1111111111111111) + Float64(0.3333333333333333 * Float64(x * cbrt(x)))) / (x ^ 2.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(Float64(x + 1.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf 46.7%

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

      1. pow1/3 43.4%

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

      2. pow-pow 90.3%

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

      3. metadata-eval 90.3%

         \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}}{{x}^{2}} \]

   5. Applied egg-rr 90.3%

      \[\leadsto \frac{-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot \color{blue}{{x}^{1.3333333333333333}}}{{x}^{2}} \]
   6. Step-by-step derivation

      1. metadata-eval 90.3%

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

      2. pow-pow 90.3%

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

      3. pow1/3 97.5%

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

      4. metadata-eval 97.5%

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

      5. pow-prod-up 97.5%

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

      6. unpow2 97.5%

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

      7. associate-*l* 97.5%

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

      8. unpow2 97.5%

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

      9. associate-*l* 97.5%

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

      10. add-cube-cbrt 99.0%

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

   7. Applied egg-rr 99.0%

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

   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 3.0%

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

      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 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in x around 0 20.0%

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

      1. fma-undefine 20.0%

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

      2. +-commutative 20.0%

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

      3. +-commutative 20.0%

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

   9. Applied egg-rr 20.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot -0.1111111111111111 + 0.3333333333333333 \cdot \left(x \cdot \sqrt[3]{x}\right)}{{x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 0.6× speedup

Math

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

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

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 0.3333333333333333 * cbrt((1.0 / pow(x, 2.0)));
	} else {
		tmp = 1.0 / (1.0 + (cbrt(x) * (cbrt(x) + cbrt((x + 1.0)))));
	}
	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 = 1.0 / (1.0 + (Math.cbrt(x) * (Math.cbrt(x) + Math.cbrt((x + 1.0)))));
	}
	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(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(Float64(x + 1.0))))));
	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[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(x + 1.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 6.4%

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

   3. Taylor expanded in x around inf 97.2%

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

   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 3.0%

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

      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 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in x around 0 20.0%

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

      1. fma-undefine 20.0%

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

      2. +-commutative 20.0%

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

      3. +-commutative 20.0%

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

   9. Applied egg-rr 20.0%

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

4. Final simplification 62.2%

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

Alternative 12: 56.7% 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 6.4%

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

   3. Taylor expanded in x around inf 97.2%

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

   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 3.0%

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

      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 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in x around 0 20.0%

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

   8. Taylor expanded in x around 0 17.7%

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

4. Final simplification 61.2%

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

Alternative 13: 5.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{x} + \left(0 - {x}^{0.3333333333333333}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (cbrt x) (- 0.0 (pow x 0.3333333333333333))))

C

double code(double x) {
	return cbrt(x) + (0.0 - pow(x, 0.3333333333333333));
}

Java

public static double code(double x) {
	return Math.cbrt(x) + (0.0 - Math.pow(x, 0.3333333333333333));
}

Julia

function code(x)
	return Float64(cbrt(x) + Float64(0.0 - (x ^ 0.3333333333333333)))
end

Wolfram

code[x_] := N[(N[Power[x, 1/3], $MachinePrecision] + N[(0.0 - N[Power[x, 0.3333333333333333], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 6.0%

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

5. Applied egg-rr 6.0%

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

6. Final simplification 6.0%

   \[\leadsto \sqrt[3]{x} + \left(0 - {x}^{0.3333333333333333}\right) \]
7. Add Preprocessing

Alternative 14: 50.2% 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 5.7%

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

3. Taylor expanded in x around inf 55.3%

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

4. Final simplification 55.3%

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

Alternative 15: 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 5.7%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 5.4%

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

   5. fabs-neg 5.4%

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

   6. unpow1/3 5.4%

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

   7. metadata-eval 5.4%

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

   8. pow-sqr 5.4%

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

   9. fabs-sqr 5.4%

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

   10. pow-sqr 5.4%

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

   11. metadata-eval 5.4%

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

   12. unpow1/3 5.4%

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

5. Simplified 5.4%

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

6. Final simplification 5.4%

   \[\leadsto 1 + \sqrt[3]{x} \]
7. 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 2024067 
(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)))