2cbrt (problem 3.3.4)

Percentage Accurate: 7.3% → 99.1%

Time: 16.9s

Alternatives: 8

Speedup: 2.0×

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: 7.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+15)
   (/
    1.0
    (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) (cbrt (pow (+ 1.0 x) 2.0))))
   (/ (* (cbrt x) 0.3333333333333333) x)))

C

double code(double x) {
	double tmp;
	if (x <= 4e+15) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+15)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+15], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e15

   1. Initial program 56.2%

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

      1. flip3-- 61.4%

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

      2. div-inv 61.4%

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

      3. rem-cube-cbrt 60.8%

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

      4. rem-cube-cbrt 98.7%

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

      5. +-commutative 98.7%

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

      6. distribute-rgt-out 98.6%

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

      7. +-commutative 98.6%

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

      8. fma-define 98.6%

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

      9. add-exp-log 98.4%

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

   4. Applied egg-rr 98.1%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. +-commutative 98.1%

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

      4. associate--l+ 98.1%

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

      5. +-inverses 98.1%

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

      6. metadata-eval 98.1%

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

      7. +-commutative 98.1%

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

      8. exp-prod 97.9%

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

   6. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. unpow-prod-down 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. pow-sqr 98.8%

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

   10. Simplified 98.8%

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

       1. sqr-pow 98.8%

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

       2. pow2 98.8%

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

       3. pow-to-exp 98.3%

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

       4. *-commutative 98.3%

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

       5. associate-/l* 98.3%

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

       6. metadata-eval 98.3%

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

       7. *-commutative 98.3%

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

       8. *-un-lft-identity 98.3%

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

       9. pow1/2 98.3%

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

       10. log-pow 98.3%

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

       11. rem-log-exp 98.3%

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

       12. metadata-eval 98.3%

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

       13. log1p-undefine 98.3%

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

       14. +-commutative 98.3%

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

       15. log-pow 98.8%

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

       16. pow1/3 98.6%

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

       17. add-exp-log 98.6%

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

       18. pow2 98.6%

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

   12. Applied egg-rr 98.8%

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

   if 4e15 < x

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf 26.4%

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

      1. +-commutative 26.4%

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

      2. fma-define 26.4%

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

   5. Simplified 26.4%

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

      1. frac-2neg 26.4%

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

      2. div-inv 26.4%

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

      3. *-commutative 26.4%

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

   7. Applied egg-rr 26.4%

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

      1. un-div-inv 26.4%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-/r* 0.0%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in x around inf 99.1%

       \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. flip3-- 7.0%

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

   2. div-inv 7.0%

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

   3. rem-cube-cbrt 6.7%

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

   4. rem-cube-cbrt 9.3%

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

   5. +-commutative 9.3%

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

   6. distribute-rgt-out 9.3%

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

   7. +-commutative 9.3%

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

   8. fma-define 9.3%

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

   9. add-exp-log 9.3%

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

4. Applied egg-rr 9.3%

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

   1. associate-*r/ 9.3%

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

   2. *-rgt-identity 9.3%

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

   3. +-commutative 9.3%

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

   4. associate--l+ 93.7%

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

   5. +-inverses 93.7%

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

   6. metadata-eval 93.7%

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

   7. +-commutative 93.7%

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

   8. exp-prod 92.6%

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

6. Simplified 92.6%

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

   1. add-sqr-sqrt 92.6%

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

   2. unpow-prod-down 94.3%

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

8. Applied egg-rr 94.3%

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

   1. pow-sqr 94.3%

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

10. Simplified 94.3%

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

    1. sqr-pow 94.3%

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

    2. pow2 94.3%

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

    3. pow-to-exp 93.7%

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

    4. *-commutative 93.7%

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

    5. associate-/l* 93.7%

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

    6. metadata-eval 93.7%

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

    7. *-commutative 93.7%

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

    8. *-un-lft-identity 93.7%

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

    9. pow1/2 93.7%

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

    10. log-pow 93.7%

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

    11. rem-log-exp 93.7%

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

    12. metadata-eval 93.7%

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

    13. log1p-undefine 93.7%

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

    14. +-commutative 93.7%

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

    15. log-pow 93.9%

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

    16. pow1/3 94.5%

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

    17. add-exp-log 98.5%

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

    18. pow2 98.5%

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

12. Applied egg-rr 98.5%

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

    1. pow2 98.5%

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

14. Applied egg-rr 98.5%

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

15. Final simplification 98.5%

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

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{t\_0}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= x 9.6e+14)
     (/ (- (+ 1.0 x) x) (+ (pow t_0 2.0) (* (cbrt x) (+ (cbrt x) t_0))))
     (/ (* (cbrt x) 0.3333333333333333) x))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if (x <= 9.6e+14) {
		tmp = ((1.0 + x) - x) / (pow(t_0, 2.0) + (cbrt(x) * (cbrt(x) + t_0)));
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if (x <= 9.6e+14) {
		tmp = ((1.0 + x) - x) / (Math.pow(t_0, 2.0) + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	} else {
		tmp = (Math.cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 9.6e+14)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64((t_0 ^ 2.0) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 9.6e+14], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.6e14

   1. Initial program 60.6%

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

      1. pow1/3 59.1%

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

      2. pow-to-exp 58.3%

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

   4. Applied egg-rr 58.3%

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

      1. exp-to-pow 59.1%

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

      2. pow1/3 60.6%

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

      3. flip3-- 66.3%

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

      4. rem-cube-cbrt 65.7%

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

      5. rem-cube-cbrt 98.8%

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

      6. pow2 98.8%

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

      7. distribute-rgt-out 98.7%

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

      8. +-commutative 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 9.6e14 < x

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf 26.7%

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

      1. +-commutative 26.7%

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

      2. fma-define 26.7%

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

   5. Simplified 26.7%

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

      1. frac-2neg 26.7%

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

      2. div-inv 26.7%

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

      3. *-commutative 26.7%

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

   7. Applied egg-rr 26.7%

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

      1. un-div-inv 26.7%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-/r* 0.0%

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

   9. Applied egg-rr 77.7%

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

   10. Taylor expanded in x around inf 99.0%

       \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4e+15)
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (cbrt (+ 1.0 x)))
     (pow (+ 1.0 x) 0.6666666666666666)))
   (/ (* (cbrt x) 0.3333333333333333) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4e+15) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), pow((1.0 + x), 0.6666666666666666));
	} else {
		tmp = (cbrt(x) * 0.3333333333333333) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4e+15)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), (Float64(1.0 + x) ^ 0.6666666666666666)));
	else
		tmp = Float64(Float64(cbrt(x) * 0.3333333333333333) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4e+15], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e15

   1. Initial program 56.2%

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

      1. flip3-- 61.4%

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

      2. div-inv 61.4%

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

      3. rem-cube-cbrt 60.8%

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

      4. rem-cube-cbrt 98.7%

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

      5. +-commutative 98.7%

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

      6. distribute-rgt-out 98.6%

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

      7. +-commutative 98.6%

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

      8. fma-define 98.6%

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

      9. add-exp-log 98.4%

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

   4. Applied egg-rr 98.1%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. +-commutative 98.1%

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

      4. associate--l+ 98.1%

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

      5. +-inverses 98.1%

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

      6. metadata-eval 98.1%

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

      7. +-commutative 98.1%

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

      8. exp-prod 97.9%

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

   6. Simplified 97.9%

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

      1. add-sqr-sqrt 97.9%

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

      2. unpow-prod-down 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. pow-sqr 98.8%

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

   10. Simplified 98.8%

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

       1. sqr-pow 98.8%

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

       2. pow2 98.8%

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

       3. pow-to-exp 98.3%

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

       4. *-commutative 98.3%

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

       5. associate-/l* 98.3%

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

       6. metadata-eval 98.3%

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

       7. *-commutative 98.3%

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

       8. *-un-lft-identity 98.3%

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

       9. pow1/2 98.3%

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

       10. log-pow 98.3%

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

       11. rem-log-exp 98.3%

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

       12. metadata-eval 98.3%

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

       13. log1p-undefine 98.3%

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

       14. +-commutative 98.3%

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

       15. log-pow 98.8%

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

       16. pow1/3 98.6%

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

       17. add-exp-log 98.6%

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

       18. pow2 98.6%

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

   12. Applied egg-rr 98.6%

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

       1. pow2 98.6%

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

       2. pow1/3 98.5%

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

       3. pow-pow 98.5%

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

       4. metadata-eval 98.5%

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

   14. Applied egg-rr 98.5%

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

   if 2.4e15 < x

   1. Initial program 4.1%

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

   3. Taylor expanded in x around inf 26.4%

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

      1. +-commutative 26.4%

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

      2. fma-define 26.4%

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

   5. Simplified 26.4%

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

      1. frac-2neg 26.4%

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

      2. div-inv 26.4%

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

      3. *-commutative 26.4%

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

   7. Applied egg-rr 26.4%

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

      1. un-div-inv 26.4%

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

      2. add-sqr-sqrt 0.0%

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

      3. associate-/r* 0.0%

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

   9. Applied egg-rr 77.6%

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

   10. Taylor expanded in x around inf 99.1%

       \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{x}}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\frac{0.3333333333333333}{x} - \frac{0.1111111111111111}{{x}^{2}}\right)}{{\left(\sqrt[3]{x}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x (- (/ 0.3333333333333333 x) (/ 0.1111111111111111 (pow x 2.0))))
  (pow (cbrt x) 2.0)))

C

double code(double x) {
	return (x * ((0.3333333333333333 / x) - (0.1111111111111111 / pow(x, 2.0)))) / pow(cbrt(x), 2.0);
}

Java

public static double code(double x) {
	return (x * ((0.3333333333333333 / x) - (0.1111111111111111 / Math.pow(x, 2.0)))) / Math.pow(Math.cbrt(x), 2.0);
}

Julia

function code(x)
	return Float64(Float64(x * Float64(Float64(0.3333333333333333 / x) - Float64(0.1111111111111111 / (x ^ 2.0)))) / (cbrt(x) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(x * N[(N[(0.3333333333333333 / x), $MachinePrecision] - N[(0.1111111111111111 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf 29.1%

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

   1. +-commutative 29.1%

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

   2. fma-define 29.1%

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

5. Simplified 29.1%

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

   1. *-un-lft-identity 29.1%

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

   2. add-cbrt-cube 18.3%

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

   3. pow-sqr 18.3%

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

   4. metadata-eval 18.3%

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

   5. cbrt-prod 28.3%

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

   6. unpow2 28.3%

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

   7. cbrt-prod 28.2%

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

   8. times-frac 28.1%

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

   9. *-commutative 28.1%

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

   10. pow2 28.1%

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

7. Applied egg-rr 28.1%

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

   1. associate-*r/ 28.2%

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

9. Simplified 77.7%

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

10. Taylor expanded in x around inf 97.3%

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

    1. associate-*r/ 97.4%

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

    2. metadata-eval 97.4%

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

    3. associate-*r/ 97.4%

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

    4. metadata-eval 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

    \[\leadsto \frac{x \cdot \left(\frac{0.3333333333333333}{x} - \frac{0.1111111111111111}{{x}^{2}}\right)}{{\left(\sqrt[3]{x}\right)}^{2}} \]
14. Add Preprocessing

Alternative 6: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.3333333333333333 + x \cdot \left(-0.1111111111111111 \cdot {x}^{-2}\right)}{{\left(\sqrt[3]{x}\right)}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+ 0.3333333333333333 (* x (* -0.1111111111111111 (pow x -2.0))))
  (pow (cbrt x) 2.0)))

C

double code(double x) {
	return (0.3333333333333333 + (x * (-0.1111111111111111 * pow(x, -2.0)))) / pow(cbrt(x), 2.0);
}

Java

public static double code(double x) {
	return (0.3333333333333333 + (x * (-0.1111111111111111 * Math.pow(x, -2.0)))) / Math.pow(Math.cbrt(x), 2.0);
}

Julia

function code(x)
	return Float64(Float64(0.3333333333333333 + Float64(x * Float64(-0.1111111111111111 * (x ^ -2.0)))) / (cbrt(x) ^ 2.0))
end

Wolfram

code[x_] := N[(N[(0.3333333333333333 + N[(x * N[(-0.1111111111111111 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf 29.1%

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

   1. +-commutative 29.1%

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

   2. fma-define 29.1%

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

5. Simplified 29.1%

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

   1. *-un-lft-identity 29.1%

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

   2. add-cbrt-cube 18.3%

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

   3. pow-sqr 18.3%

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

   4. metadata-eval 18.3%

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

   5. cbrt-prod 28.3%

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

   6. unpow2 28.3%

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

   7. cbrt-prod 28.2%

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

   8. times-frac 28.1%

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

   9. *-commutative 28.1%

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

   10. pow2 28.1%

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

7. Applied egg-rr 28.1%

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

   1. associate-*r/ 28.2%

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

9. Simplified 77.7%

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

10. Taylor expanded in x around inf 97.3%

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

    1. sub-neg 97.3%

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

    2. distribute-lft-in 97.3%

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

    3. *-commutative 97.3%

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

    4. associate-*r* 97.3%

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

    5. rgt-mult-inverse 97.4%

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

    6. metadata-eval 97.4%

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

    7. distribute-lft-neg-in 97.4%

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

    8. metadata-eval 97.4%

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

    9. unpow2 97.4%

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

    10. associate-/r* 97.4%

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

    11. *-rgt-identity 97.4%

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

    12. associate-*r/ 97.4%

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

    13. unpow-1 97.4%

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

    14. unpow-1 97.4%

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

    15. pow-sqr 97.4%

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

    16. metadata-eval 97.4%

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

12. Simplified 97.4%

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

13. Final simplification 97.4%

    \[\leadsto \frac{0.3333333333333333 + x \cdot \left(-0.1111111111111111 \cdot {x}^{-2}\right)}{{\left(\sqrt[3]{x}\right)}^{2}} \]
14. Add Preprocessing

Alternative 7: 96.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

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

3. Taylor expanded in x around inf 29.1%

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

   1. +-commutative 29.1%

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

   2. fma-define 29.1%

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

5. Simplified 29.1%

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

   1. frac-2neg 29.1%

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

   2. div-inv 29.0%

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

   3. *-commutative 29.0%

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

7. Applied egg-rr 29.0%

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

   1. un-div-inv 29.1%

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

   2. add-sqr-sqrt 0.0%

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

   3. associate-/r* 0.0%

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

9. Applied egg-rr 77.6%

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

10. Taylor expanded in x around inf 97.1%

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

11. Final simplification 97.1%

    \[\leadsto \frac{\sqrt[3]{x} \cdot 0.3333333333333333}{x} \]
12. Add Preprocessing

Alternative 8: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.8%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 5.5%

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

   5. fabs-neg 5.5%

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

   6. unpow1/3 5.5%

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

   7. metadata-eval 5.5%

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

   8. pow-sqr 5.5%

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

   9. fabs-sqr 5.5%

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

   10. pow-sqr 5.5%

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

   11. metadata-eval 5.5%

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

   12. unpow1/3 5.5%

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

5. Simplified 5.5%

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

6. Final simplification 5.5%

   \[\leadsto 1 + \sqrt[3]{x} \]
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 2024060 
(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)))