2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 98.5%

Time: 9.9s

Alternatives: 11

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: 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: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. flip3-- 7.5%

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

   2. div-inv 7.5%

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

   3. rem-cube-cbrt 7.7%

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

   4. rem-cube-cbrt 10.0%

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

   5. +-commutative 10.0%

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

   6. distribute-rgt-out 10.0%

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

   7. +-commutative 10.0%

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

   8. fma-define 10.0%

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

   9. add-exp-log 10.0%

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

4. Applied egg-rr 10.0%

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

   1. associate-*r/ 10.0%

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

   2. *-rgt-identity 10.0%

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

   3. +-commutative 10.0%

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

   4. associate--l+ 93.5%

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

   5. +-commutative 93.5%

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

   6. +-commutative 93.5%

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

6. Simplified 93.5%

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

   1. *-commutative 93.5%

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

   2. log1p-undefine 93.5%

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

   3. +-commutative 93.5%

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

   4. exp-to-pow 93.3%

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

   5. metadata-eval 93.3%

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

   6. pow-prod-up 93.3%

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

   7. pow1/3 94.7%

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

   8. pow1/3 98.5%

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

   9. pow2 98.5%

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

   10. +-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. pow1/3 93.3%

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

   2. add-sqr-sqrt 93.3%

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

   3. unpow-prod-down 93.3%

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

10. Applied egg-rr 93.3%

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

    1. unpow1/3 94.7%

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

    2. unpow1/3 98.6%

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

12. Simplified 98.6%

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

13. Taylor expanded in x around 0 98.6%

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

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(x - x\right)\\ t_1 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_1, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} \cdot 2, {t\_1}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (- x x))) (t_1 (cbrt (+ 1.0 x))))
   (if (<= x 4.6e+15)
     (/ t_0 (fma (cbrt x) (+ (cbrt x) t_1) (pow (+ 1.0 x) 0.6666666666666666)))
     (/ t_0 (fma (cbrt x) (* (cbrt x) 2.0) (pow t_1 2.0))))))

C

double code(double x) {
	double t_0 = 1.0 + (x - x);
	double t_1 = cbrt((1.0 + x));
	double tmp;
	if (x <= 4.6e+15) {
		tmp = t_0 / fma(cbrt(x), (cbrt(x) + t_1), pow((1.0 + x), 0.6666666666666666));
	} else {
		tmp = t_0 / fma(cbrt(x), (cbrt(x) * 2.0), pow(t_1, 2.0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x - x))
	t_1 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 4.6e+15)
		tmp = Float64(t_0 / fma(cbrt(x), Float64(cbrt(x) + t_1), (Float64(1.0 + x) ^ 0.6666666666666666)));
	else
		tmp = Float64(t_0 / fma(cbrt(x), Float64(cbrt(x) * 2.0), (t_1 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 4.6e+15], N[(t$95$0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$1), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] * 2.0), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.6e15

   1. Initial program 58.7%

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

      1. flip3-- 58.6%

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

      2. div-inv 58.6%

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

      3. rem-cube-cbrt 71.3%

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

      4. rem-cube-cbrt 98.5%

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

      5. +-commutative 98.5%

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

      6. distribute-rgt-out 98.5%

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

      7. +-commutative 98.5%

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

      8. fma-define 98.5%

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

      9. add-exp-log 98.2%

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

   4. Applied egg-rr 98.0%

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

      1. associate-*r/ 98.0%

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

      2. *-rgt-identity 98.0%

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

      3. +-commutative 98.0%

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

      4. associate--l+ 98.0%

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

      5. +-commutative 98.0%

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

      6. +-commutative 98.0%

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

   6. Simplified 98.0%

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

      1. *-commutative 98.0%

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

      2. exp-prod 98.2%

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

      3. log1p-undefine 98.4%

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

      4. add-exp-log 98.6%

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

   8. Applied egg-rr 98.6%

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

   if 4.6e15 < x

   1. Initial program 4.1%

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

      1. flip3-- 4.1%

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

      2. div-inv 4.1%

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

      3. rem-cube-cbrt 3.5%

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

      4. rem-cube-cbrt 4.1%

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

      5. +-commutative 4.1%

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

      6. distribute-rgt-out 4.1%

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

      7. +-commutative 4.1%

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

      8. fma-define 4.1%

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

      9. add-exp-log 4.1%

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

   4. Applied egg-rr 4.1%

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

      1. associate-*r/ 4.1%

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

      2. *-rgt-identity 4.1%

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

      3. +-commutative 4.1%

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

      4. associate--l+ 93.2%

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

      5. +-commutative 93.2%

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

      6. +-commutative 93.2%

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

   6. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. log1p-undefine 93.2%

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

      3. +-commutative 93.2%

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

      4. exp-to-pow 92.9%

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

      5. metadata-eval 92.9%

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

      6. pow-prod-up 92.9%

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

      7. pow1/3 94.4%

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

      8. pow1/3 98.5%

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

      9. pow2 98.5%

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

      10. +-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 98.5%

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

   11. Simplified 98.5%

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

Alternative 3: 98.4% 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 7.5%

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

   1. flip3-- 7.5%

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

   2. div-inv 7.5%

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

   3. rem-cube-cbrt 7.7%

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

   4. rem-cube-cbrt 10.0%

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

   5. +-commutative 10.0%

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

   6. distribute-rgt-out 10.0%

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

   7. +-commutative 10.0%

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

   8. fma-define 10.0%

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

   9. add-exp-log 10.0%

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

4. Applied egg-rr 10.0%

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

   1. associate-*r/ 10.0%

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

   2. *-rgt-identity 10.0%

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

   3. +-commutative 10.0%

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

   4. associate--l+ 93.5%

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

   5. +-commutative 93.5%

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

   6. +-commutative 93.5%

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

6. Simplified 93.5%

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

   1. *-commutative 93.5%

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

   2. log1p-undefine 93.5%

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

   3. +-commutative 93.5%

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

   4. exp-to-pow 93.3%

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

   5. metadata-eval 93.3%

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

   6. pow-prod-up 93.3%

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

   7. pow1/3 94.7%

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

   8. pow1/3 98.5%

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

   9. pow2 98.5%

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

   10. +-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. pow1/3 93.3%

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

   2. add-sqr-sqrt 93.3%

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

   3. unpow-prod-down 93.3%

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

10. Applied egg-rr 93.3%

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

    1. unpow1/3 94.7%

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

    2. unpow1/3 98.6%

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

12. Simplified 98.6%

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

    1. fma-undefine 98.5%

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

    2. +-commutative 98.5%

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

    3. cbrt-unprod 98.6%

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

    4. add-sqr-sqrt 98.5%

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

14. Applied egg-rr 98.5%

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

Alternative 4: 96.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. flip3-- 7.5%

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

   2. div-inv 7.5%

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

   3. rem-cube-cbrt 7.7%

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

   4. rem-cube-cbrt 10.0%

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

   5. +-commutative 10.0%

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

   6. distribute-rgt-out 10.0%

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

   7. +-commutative 10.0%

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

   8. fma-define 10.0%

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

   9. add-exp-log 10.0%

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

4. Applied egg-rr 10.0%

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

   1. associate-*r/ 10.0%

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

   2. *-rgt-identity 10.0%

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

   3. +-commutative 10.0%

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

   4. associate--l+ 93.5%

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

   5. +-commutative 93.5%

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

   6. +-commutative 93.5%

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

6. Simplified 93.5%

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

   1. *-commutative 93.5%

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

   2. log1p-undefine 93.5%

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

   3. +-commutative 93.5%

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

   4. exp-to-pow 93.3%

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

   5. metadata-eval 93.3%

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

   6. pow-prod-up 93.3%

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

   7. pow1/3 94.7%

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

   8. pow1/3 98.5%

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

   9. pow2 98.5%

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

   10. +-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 96.2%

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

11. Simplified 96.2%

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

Alternative 5: 92.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{{x}^{-2} \cdot 0.037037037037037035}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.1e+154)
   (cbrt (* (pow x -2.0) 0.037037037037037035))
   (* 0.3333333333333333 (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 4.1e+154) {
		tmp = cbrt((pow(x, -2.0) * 0.037037037037037035));
	} else {
		tmp = 0.3333333333333333 * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.1e+154) {
		tmp = Math.cbrt((Math.pow(x, -2.0) * 0.037037037037037035));
	} else {
		tmp = 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.1e+154)
		tmp = cbrt(Float64((x ^ -2.0) * 0.037037037037037035));
	else
		tmp = Float64(0.3333333333333333 * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.1e+154], N[Power[N[(N[Power[x, -2.0], $MachinePrecision] * 0.037037037037037035), $MachinePrecision], 1/3], $MachinePrecision], N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e154

   1. Initial program 9.8%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. add-sqr-sqrt 93.4%

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

      2. sqrt-unprod 93.7%

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

      3. cbrt-unprod 49.6%

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

      4. pow-flip 49.6%

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

      5. pow-flip 49.5%

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

      6. pow-sqr 49.5%

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

      7. metadata-eval 49.5%

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

      8. metadata-eval 49.5%

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

   5. Applied egg-rr 49.5%

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

      1. metadata-eval 49.5%

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

      2. pow-sqr 49.5%

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

      3. cbrt-prod 94.3%

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

      4. sqrt-unprod 94.0%

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

      5. add-sqr-sqrt 94.3%

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

      6. pow1 94.3%

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

   7. Applied egg-rr 94.3%

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

      1. unpow1 94.3%

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

      2. rem-cbrt-cube 93.8%

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

      3. *-commutative 93.8%

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

      4. cube-prod 93.7%

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

      5. rem-cube-cbrt 94.2%

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

      6. metadata-eval 94.3%

         \[\leadsto \sqrt[3]{{x}^{-2} \cdot \color{blue}{0.037037037037037035}} \]

   9. Simplified 94.3%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{-2} \cdot 0.037037037037037035}} \]

   if 4.1e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf 4.7%

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

      1. pow1/3 4.7%

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

      2. pow-flip 8.6%

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

      3. pow-pow 89.2%

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

      4. metadata-eval 89.2%

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

      5. metadata-eval 89.2%

         \[\leadsto 0.3333333333333333 \cdot {x}^{\color{blue}{-0.6666666666666666}} \]

   5. Applied egg-rr 89.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{x}^{-0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 53.7%

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

   1. cbrt-div 53.8%

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

   2. metadata-eval 53.8%

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

   3. unpow2 53.8%

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

   4. cbrt-prod 96.0%

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

   5. pow2 96.0%

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

5. Applied egg-rr 96.0%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
6. Add Preprocessing

Alternative 7: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 53.7%

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

   1. cbrt-div 53.8%

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

   2. metadata-eval 53.8%

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

   3. un-div-inv 53.8%

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

   4. unpow2 53.8%

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

   5. cbrt-prod 95.9%

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

   6. pow2 95.9%

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

5. Applied egg-rr 95.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{{\left(\sqrt[3]{x}\right)}^{2}}} \]
6. Add Preprocessing

Alternative 8: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{+155}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.6666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.05e+155)
   (* 0.3333333333333333 (cbrt (* (/ 1.0 x) (/ 1.0 x))))
   (* 0.3333333333333333 (pow x -0.6666666666666666))))

C

double code(double x) {
	double tmp;
	if (x <= 2.05e+155) {
		tmp = 0.3333333333333333 * cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * pow(x, -0.6666666666666666);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.05e+155) {
		tmp = 0.3333333333333333 * Math.cbrt(((1.0 / x) * (1.0 / x)));
	} else {
		tmp = 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.05e+155)
		tmp = Float64(0.3333333333333333 * cbrt(Float64(Float64(1.0 / x) * Float64(1.0 / x))));
	else
		tmp = Float64(0.3333333333333333 * (x ^ -0.6666666666666666));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.05e+155], N[(0.3333333333333333 * N[Power[N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0499999999999999e155

   1. Initial program 9.8%

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

   3. Taylor expanded in x around inf 93.7%

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

      1. add-sqr-sqrt 93.7%

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

      2. sqrt-div 93.6%

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

      3. metadata-eval 93.6%

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

      4. sqrt-pow1 93.6%

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

      5. metadata-eval 93.6%

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

      6. pow1 93.6%

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

      7. sqrt-div 93.6%

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

      8. metadata-eval 93.6%

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

      9. sqrt-pow1 94.3%

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

      10. metadata-eval 94.3%

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

      11. pow1 94.3%

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

   5. Applied egg-rr 94.3%

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

   if 2.0499999999999999e155 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf 4.7%

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

      1. pow1/3 4.7%

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

      2. pow-flip 8.6%

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

      3. pow-pow 89.2%

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

      4. metadata-eval 89.2%

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

      5. metadata-eval 89.2%

         \[\leadsto 0.3333333333333333 \cdot {x}^{\color{blue}{-0.6666666666666666}} \]

   5. Applied egg-rr 89.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{x}^{-0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 92.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 9.9%

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

   3. Taylor expanded in x around inf 94.3%

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

      1. unpow2 94.3%

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

   5. Applied egg-rr 94.3%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

   3. Taylor expanded in x around inf 4.7%

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

      1. pow1/3 4.7%

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

      2. pow-flip 9.3%

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

      3. pow-pow 89.2%

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

      4. metadata-eval 89.2%

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

      5. metadata-eval 89.2%

         \[\leadsto 0.3333333333333333 \cdot {x}^{\color{blue}{-0.6666666666666666}} \]

   5. Applied egg-rr 89.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{{x}^{-0.6666666666666666}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.3333333333333333 (pow x -0.6666666666666666)))

C

double code(double x) {
	return 0.3333333333333333 * pow(x, -0.6666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0 * (x ** (-0.6666666666666666d0))
end function

Java

public static double code(double x) {
	return 0.3333333333333333 * Math.pow(x, -0.6666666666666666);
}

Python

def code(x):
	return 0.3333333333333333 * math.pow(x, -0.6666666666666666)

Julia

function code(x)
	return Float64(0.3333333333333333 * (x ^ -0.6666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333 * (x ^ -0.6666666666666666);
end

Wolfram

code[x_] := N[(0.3333333333333333 * N[Power[x, -0.6666666666666666], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 53.7%

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

   1. pow1/3 50.3%

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

   2. pow-flip 52.4%

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

   3. pow-pow 88.6%

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

   4. metadata-eval 88.6%

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

   5. metadata-eval 88.6%

      \[\leadsto 0.3333333333333333 \cdot {x}^{\color{blue}{-0.6666666666666666}} \]

5. Applied egg-rr 88.6%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{x}^{-0.6666666666666666}} \]
6. Add Preprocessing

Alternative 11: 5.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

code[x_] := N[Power[x, 1/3], $MachinePrecision]

Derivation

1. Initial program 7.5%

   \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
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 5.6%

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

   5. fabs-neg 5.6%

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

   6. unpow1/3 5.6%

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

   7. metadata-eval 5.6%

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

   8. pow-sqr 5.6%

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

   9. fabs-sqr 5.6%

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

   10. pow-sqr 5.6%

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

   11. metadata-eval 5.6%

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

   12. unpow1/3 5.6%

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

5. Simplified 5.6%

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

6. Taylor expanded in x around inf 5.6%

   \[\leadsto \color{blue}{\sqrt[3]{x}} \]
7. Add Preprocessing

Developer Target 1: 98.4% 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 2024186 
(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)))