2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.9%

Time: 11.8s

Alternatives: 9

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: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+43}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(2, \sqrt[3]{\frac{1}{x}}, \sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+43)
   (/
    1.0
    (+
     (pow (cbrt (+ 1.0 x)) 2.0)
     (+ (pow (cbrt x) 2.0) (cbrt (* x (+ 1.0 x))))))
   (/
    (/ 1.0 x)
    (fma 2.0 (cbrt (/ 1.0 x)) (cbrt (+ (/ 1.0 x) (/ 2.0 (pow x 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+43) {
		tmp = 1.0 / (pow(cbrt((1.0 + x)), 2.0) + (pow(cbrt(x), 2.0) + cbrt((x * (1.0 + x)))));
	} else {
		tmp = (1.0 / x) / fma(2.0, cbrt((1.0 / x)), cbrt(((1.0 / x) + (2.0 / pow(x, 2.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+43)
		tmp = Float64(1.0 / Float64((cbrt(Float64(1.0 + x)) ^ 2.0) + Float64((cbrt(x) ^ 2.0) + cbrt(Float64(x * Float64(1.0 + x))))));
	else
		tmp = Float64(Float64(1.0 / x) / fma(2.0, cbrt(Float64(1.0 / x)), cbrt(Float64(Float64(1.0 / x) + Float64(2.0 / (x ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+43], N[(1.0 / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(2.0 * N[Power[N[(1.0 / x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / x), $MachinePrecision] + N[(2.0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000001e43

   1. Initial program 30.0%

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

      1. add-exp-log 30.4%

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

   4. Applied egg-rr 30.4%

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

      1. rem-exp-log 30.0%

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

      2. flip3-- 30.0%

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

      3. +-commutative 30.0%

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

      4. rem-cube-cbrt 30.7%

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

      5. rem-cube-cbrt 52.5%

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

      6. associate-+r- 98.9%

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

      7. +-inverses 98.9%

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

      8. metadata-eval 98.9%

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

      9. +-commutative 98.9%

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

      10. +-commutative 98.9%

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

      11. unpow2 98.9%

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

      12. pow2 98.9%

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

      13. +-commutative 98.9%

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

      14. cbrt-unprod 99.0%

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

   6. Applied egg-rr 99.0%

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

      1. +-commutative 99.0%

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

      2. *-commutative 99.0%

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

      3. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   if 1.00000000000000001e43 < x

   1. Initial program 4.3%

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

      1. flip3-- 4.3%

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

      2. div-inv 4.3%

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

      3. rem-cube-cbrt 3.4%

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

      4. rem-cube-cbrt 4.3%

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

      5. +-commutative 4.3%

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

      6. distribute-rgt-out 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-define 4.3%

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

      9. add-exp-log 4.3%

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

   4. Applied egg-rr 4.3%

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

      1. associate-*r/ 4.3%

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

      2. *-rgt-identity 4.3%

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

      3. +-commutative 4.3%

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

      4. associate--l+ 92.7%

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

      5. +-commutative 92.7%

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

      6. +-commutative 92.7%

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

   6. Simplified 92.7%

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

      1. *-commutative 92.7%

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

      2. exp-prod 92.8%

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

      3. log1p-undefine 92.8%

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

      4. add-exp-log 92.4%

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

      5. +-commutative 92.4%

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

      6. metadata-eval 92.4%

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

      7. pow-sqr 92.4%

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

      8. pow1/3 93.9%

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

      9. pow1/3 98.5%

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

      10. pow2 98.5%

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

      11. +-commutative 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in x around inf 20.6%

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

   10. Taylor expanded in x around inf 98.7%

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

       1. associate-/r* 98.8%

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

       2. +-commutative 98.8%

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

       3. fma-define 98.8%

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

       4. associate-*r/ 98.8%

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

       5. metadata-eval 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+43}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left({\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(2, \sqrt[3]{\frac{1}{x}}, \sqrt[3]{\frac{1}{x} + \frac{2}{{x}^{2}}}\right)}\\ \end{array} \]
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 8.0%

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

   1. flip3-- 8.0%

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

   2. div-inv 8.0%

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

   3. rem-cube-cbrt 7.3%

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

   4. rem-cube-cbrt 11.3%

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

   5. +-commutative 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. +-commutative 11.3%

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

   8. fma-define 11.3%

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

   9. add-exp-log 11.2%

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

4. Applied egg-rr 11.2%

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

   1. associate-*r/ 11.2%

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

   2. *-rgt-identity 11.2%

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

   3. +-commutative 11.2%

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

   4. associate--l+ 93.3%

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

   5. +-commutative 93.3%

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

   6. +-commutative 93.3%

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

6. Simplified 93.3%

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

   1. *-commutative 93.3%

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

   2. exp-prod 93.5%

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

   3. log1p-undefine 93.5%

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

   4. add-exp-log 93.1%

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

   5. +-commutative 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow-sqr 93.1%

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

   8. pow1/3 94.5%

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

   9. pow1/3 98.5%

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

   10. pow2 98.5%

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

   11. +-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around 0 98.5%

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

Alternative 3: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. flip3-- 8.0%

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

   2. div-inv 8.0%

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

   3. rem-cube-cbrt 7.3%

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

   4. rem-cube-cbrt 11.3%

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

   5. +-commutative 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. +-commutative 11.3%

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

   8. fma-define 11.3%

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

   9. add-exp-log 11.2%

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

4. Applied egg-rr 11.2%

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

   1. associate-*r/ 11.2%

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

   2. *-rgt-identity 11.2%

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

   3. +-commutative 11.2%

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

   4. associate--l+ 93.3%

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

   5. +-commutative 93.3%

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

   6. +-commutative 93.3%

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

6. Simplified 93.3%

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

   1. *-commutative 93.3%

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

   2. exp-prod 93.5%

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

   3. log1p-undefine 93.5%

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

   4. add-exp-log 93.1%

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

   5. +-commutative 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow-sqr 93.1%

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

   8. pow1/3 94.5%

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

   9. pow1/3 98.5%

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

   10. pow2 98.5%

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

   11. +-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

   1. add-cube-cbrt 98.4%

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

   2. pow3 98.4%

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

10. Applied egg-rr 98.4%

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

    1. fma-undefine 98.5%

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

    2. rem-cube-cbrt 98.5%

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

12. Applied egg-rr 98.5%

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

13. Final simplification 98.5%

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

Alternative 4: 97.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. flip3-- 8.0%

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

   2. div-inv 8.0%

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

   3. rem-cube-cbrt 7.3%

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

   4. rem-cube-cbrt 11.3%

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

   5. +-commutative 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. +-commutative 11.3%

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

   8. fma-define 11.3%

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

   9. add-exp-log 11.2%

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

4. Applied egg-rr 11.2%

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

   1. associate-*r/ 11.2%

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

   2. *-rgt-identity 11.2%

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

   3. +-commutative 11.2%

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

   4. associate--l+ 93.3%

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

   5. +-commutative 93.3%

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

   6. +-commutative 93.3%

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

6. Simplified 93.3%

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

   1. *-commutative 93.3%

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

   2. exp-prod 93.5%

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

   3. log1p-undefine 93.5%

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

   4. add-exp-log 93.1%

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

   5. +-commutative 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow-sqr 93.1%

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

   8. pow1/3 94.5%

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

   9. pow1/3 98.5%

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

   10. pow2 98.5%

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

   11. +-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around inf 29.4%

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

10. Taylor expanded in x around inf 96.3%

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

    1. associate-/r* 96.4%

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

    2. +-commutative 96.4%

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

    3. fma-define 96.4%

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

    4. associate-*r/ 96.4%

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

    5. metadata-eval 96.4%

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

12. Simplified 96.4%

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

Alternative 5: 97.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. flip3-- 8.0%

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

   2. div-inv 8.0%

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

   3. rem-cube-cbrt 7.3%

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

   4. rem-cube-cbrt 11.3%

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

   5. +-commutative 11.3%

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

   6. distribute-rgt-out 11.3%

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

   7. +-commutative 11.3%

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

   8. fma-define 11.3%

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

   9. add-exp-log 11.2%

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

4. Applied egg-rr 11.2%

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

   1. associate-*r/ 11.2%

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

   2. *-rgt-identity 11.2%

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

   3. +-commutative 11.2%

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

   4. associate--l+ 93.3%

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

   5. +-commutative 93.3%

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

   6. +-commutative 93.3%

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

6. Simplified 93.3%

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

   1. *-commutative 93.3%

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

   2. exp-prod 93.5%

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

   3. log1p-undefine 93.5%

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

   4. add-exp-log 93.1%

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

   5. +-commutative 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow-sqr 93.1%

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

   8. pow1/3 94.5%

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

   9. pow1/3 98.5%

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

   10. pow2 98.5%

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

   11. +-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around inf 29.4%

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

10. Taylor expanded in x around inf 96.3%

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

    1. associate-*r/ 96.3%

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

    2. metadata-eval 96.3%

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

12. Simplified 96.3%

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

Alternative 6: 50.5% 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 8.0%

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

   1. add-exp-log 7.0%

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

   2. +-commutative 7.0%

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

   3. log1p-define 7.0%

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

4. Applied egg-rr 7.0%

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

   1. log1p-undefine 7.0%

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

   2. add-exp-log 8.0%

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

   3. +-commutative 8.0%

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

   4. add-exp-log 7.4%

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

   5. pow1/3 6.0%

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

   6. log-pow 6.2%

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

   7. +-commutative 6.2%

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

   8. log1p-undefine 6.2%

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

6. Applied egg-rr 6.2%

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

8. Simplified 6.3%

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

9. Taylor expanded in x around inf 50.6%

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

Alternative 7: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

3. Final simplification 8.0%

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

Alternative 8: 5.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. sub-neg 8.0%

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

4. Applied egg-rr 8.0%

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

   1. rem-square-sqrt 0.0%

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

   2. fabs-sqr 0.0%

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

   3. rem-square-sqrt 5.4%

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

   4. fabs-neg 5.4%

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

   5. unpow1/3 5.4%

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

   6. metadata-eval 5.4%

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

   7. pow-sqr 5.4%

      \[\leadsto \sqrt[3]{x + 1} + \left|\color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}}\right| \]

   8. fabs-sqr 5.4%

      \[\leadsto \sqrt[3]{x + 1} + \color{blue}{{x}^{0.16666666666666666} \cdot {x}^{0.16666666666666666}} \]

   9. pow-sqr 5.4%

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

   10. metadata-eval 5.4%

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

   11. unpow1/3 5.4%

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

   12. +-commutative 5.4%

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

6. Simplified 5.4%

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

7. Final simplification 5.4%

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

Alternative 9: 4.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.0%

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

   1. add-exp-log 7.0%

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

   2. +-commutative 7.0%

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

   3. log1p-define 7.0%

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

4. Applied egg-rr 7.0%

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

5. Taylor expanded in x around 0 4.3%

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

7. Simplified 4.3%

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

Developer Target 1: 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 2024179 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

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