2cbrt (problem 3.3.4)

Percentage Accurate: 7.1% → 98.6%

Time: 11.7s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 7.1% 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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ t_1 := \sqrt[3]{t\_0}\\ \frac{1 + \left(x - x\right)}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t\_1 \cdot t\_1, {\left(\sqrt[3]{{t\_0}^{2}}\right)}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))) (t_1 (cbrt t_0)))
   (/
    (+ 1.0 (- x x))
    (fma (cbrt x) (+ (cbrt x) (* t_1 t_1)) (pow (cbrt (pow t_0 2.0)) 2.0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double t_1 = cbrt(t_0);
	return (1.0 + (x - x)) / fma(cbrt(x), (cbrt(x) + (t_1 * t_1)), pow(cbrt(pow(t_0, 2.0)), 2.0));
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	t_1 = cbrt(t_0)
	return Float64(Float64(1.0 + Float64(x - x)) / fma(cbrt(x), Float64(cbrt(x) + Float64(t_1 * t_1)), (cbrt((t_0 ^ 2.0)) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. flip3-- 6.7%

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

   2. div-inv 6.7%

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

   3. rem-cube-cbrt 5.9%

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

   4. rem-cube-cbrt 7.6%

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

   5. +-commutative 7.6%

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

   6. distribute-rgt-out 7.6%

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

   7. +-commutative 7.6%

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

   8. fma-define 7.6%

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

   9. add-exp-log 7.6%

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

4. Applied egg-rr 7.6%

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

   1. associate-*r/ 7.6%

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

   2. *-rgt-identity 7.6%

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

   3. +-commutative 7.6%

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

   4. associate--l+ 93.0%

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

   5. +-commutative 93.0%

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

   6. +-commutative 93.0%

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

6. Simplified 93.0%

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

   1. *-commutative 93.0%

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

   2. log1p-undefine 93.0%

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

   3. exp-to-pow 92.8%

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

   4. metadata-eval 92.8%

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

   5. pow-prod-up 92.8%

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

   6. +-commutative 92.8%

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

   7. pow1/3 94.3%

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

   8. +-commutative 94.3%

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

   9. pow1/3 98.3%

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

   10. pow2 98.3%

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

   11. +-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

   1. pow1/3 94.3%

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

   2. add-sqr-sqrt 94.3%

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

   3. unpow-prod-down 94.3%

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

10. Applied egg-rr 94.3%

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

    1. unpow1/3 95.7%

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

    2. unpow1/3 98.4%

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

12. Simplified 98.4%

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

    1. add-sqr-sqrt 98.5%

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

    2. pow2 98.5%

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

14. Applied egg-rr 98.5%

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

Alternative 2: 98.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 6.6%

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

   1. flip3-- 6.7%

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

   2. div-inv 6.7%

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

   3. rem-cube-cbrt 5.9%

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

   4. rem-cube-cbrt 7.6%

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

   5. +-commutative 7.6%

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

   6. distribute-rgt-out 7.6%

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

   7. +-commutative 7.6%

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

   8. fma-define 7.6%

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

   9. add-exp-log 7.6%

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

4. Applied egg-rr 7.6%

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

   1. associate-*r/ 7.6%

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

   2. *-rgt-identity 7.6%

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

   3. +-commutative 7.6%

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

   4. associate--l+ 93.0%

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

   5. +-commutative 93.0%

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

   6. +-commutative 93.0%

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

6. Simplified 93.0%

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

   1. *-commutative 93.0%

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

   2. log1p-undefine 93.0%

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

   3. exp-to-pow 92.8%

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

   4. metadata-eval 92.8%

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

   5. pow-prod-up 92.8%

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

   6. +-commutative 92.8%

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

   7. pow1/3 94.3%

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

   8. +-commutative 94.3%

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

   9. pow1/3 98.3%

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

   10. pow2 98.3%

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

   11. +-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

   1. pow1/3 94.3%

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

   2. add-sqr-sqrt 94.3%

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

   3. unpow-prod-down 94.3%

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

10. Applied egg-rr 94.3%

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

    1. unpow1/3 95.7%

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

    2. unpow1/3 98.4%

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

12. Simplified 98.4%

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

13. Taylor expanded in x around 0 98.4%

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

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} \cdot 2, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/
    1.0
    (fma (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))) (cbrt (pow (+ 1.0 x) 2.0))))
   (/
    1.0
    (fma (cbrt x) (* (cbrt x) 2.0) (exp (* 0.6666666666666666 (log1p x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), cbrt(pow((1.0 + x), 2.0)));
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) * 2.0), exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), cbrt((Float64(1.0 + x) ^ 2.0))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) * 2.0), exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] * 2.0), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.7%

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

      1. flip3-- 8.9%

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

      2. div-inv 8.9%

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

      3. rem-cube-cbrt 8.9%

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

      4. rem-cube-cbrt 10.7%

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

      5. +-commutative 10.7%

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

      6. distribute-rgt-out 10.7%

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

      7. +-commutative 10.7%

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

      8. fma-define 10.7%

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

      9. add-exp-log 10.7%

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

   4. Applied egg-rr 10.7%

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

      1. associate-*r/ 10.7%

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

      2. *-rgt-identity 10.7%

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

      3. +-commutative 10.7%

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

      4. associate--l+ 94.5%

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

      5. +-commutative 94.5%

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

      6. +-commutative 94.5%

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

   6. Simplified 94.5%

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

      1. *-commutative 94.5%

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

      2. log1p-undefine 94.5%

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

      3. exp-to-pow 94.3%

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

      4. metadata-eval 94.3%

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

      5. pow-prod-up 94.3%

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

      6. +-commutative 94.3%

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

      7. pow1/3 95.7%

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

      8. +-commutative 95.7%

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

      9. pow1/3 98.3%

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

      10. pow2 98.3%

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

      11. +-commutative 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. *-un-lft-identity 98.0%

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

       2. +-inverses 98.0%

          \[\leadsto 1 \cdot \frac{1 + \color{blue}{0}}{\mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{x}}\right)}^{3}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

       3. metadata-eval 98.0%

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

       4. rem-cube-cbrt 98.3%

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

       5. unpow2 98.3%

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

       6. cbrt-unprod 98.7%

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

       7. pow2 98.7%

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

   12. Applied egg-rr 98.7%

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

       1. *-lft-identity 98.7%

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

   14. Simplified 98.7%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-commutative 91.7%

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

      6. +-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in x around 0 91.7%

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

   8. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

   10. Simplified 91.7%

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

Alternative 4: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.6%

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

   1. flip3-- 6.7%

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

   2. div-inv 6.7%

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

   3. rem-cube-cbrt 5.9%

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

   4. rem-cube-cbrt 7.6%

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

   5. +-commutative 7.6%

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

   6. distribute-rgt-out 7.6%

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

   7. +-commutative 7.6%

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

   8. fma-define 7.6%

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

   9. add-exp-log 7.6%

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

4. Applied egg-rr 7.6%

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

   1. associate-*r/ 7.6%

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

   2. *-rgt-identity 7.6%

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

   3. +-commutative 7.6%

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

   4. associate--l+ 93.0%

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

   5. +-commutative 93.0%

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

   6. +-commutative 93.0%

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

6. Simplified 93.0%

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

   1. *-commutative 93.0%

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

   2. log1p-undefine 93.0%

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

   3. exp-to-pow 92.8%

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

   4. metadata-eval 92.8%

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

   5. pow-prod-up 92.8%

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

   6. +-commutative 92.8%

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

   7. pow1/3 94.3%

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

   8. +-commutative 94.3%

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

   9. pow1/3 98.3%

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

   10. pow2 98.3%

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

   11. +-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

9. Taylor expanded in x around 0 98.3%

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

Alternative 5: 95.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} \cdot 2, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154)
   (/
    (+ 1.0 (- x x))
    (+ (cbrt (pow (+ 1.0 x) 2.0)) (* (cbrt x) (+ (cbrt x) (cbrt (+ 1.0 x))))))
   (/
    1.0
    (fma (cbrt x) (* (cbrt x) 2.0) (exp (* 0.6666666666666666 (log1p x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = (1.0 + (x - x)) / (cbrt(pow((1.0 + x), 2.0)) + (cbrt(x) * (cbrt(x) + cbrt((1.0 + x)))));
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) * 2.0), exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(cbrt((Float64(1.0 + x) ^ 2.0)) + Float64(cbrt(x) * Float64(cbrt(x) + cbrt(Float64(1.0 + x))))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) * 2.0), exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] * 2.0), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.7%

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

      1. flip3-- 8.9%

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

      2. div-inv 8.9%

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

      3. rem-cube-cbrt 8.9%

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

      4. rem-cube-cbrt 10.7%

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

      5. +-commutative 10.7%

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

      6. distribute-rgt-out 10.7%

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

      7. +-commutative 10.7%

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

      8. fma-define 10.7%

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

      9. add-exp-log 10.7%

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

   4. Applied egg-rr 10.7%

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

      1. associate-*r/ 10.7%

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

      2. *-rgt-identity 10.7%

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

      3. +-commutative 10.7%

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

      4. associate--l+ 94.5%

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

      5. +-commutative 94.5%

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

      6. +-commutative 94.5%

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

   6. Simplified 94.5%

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

      1. *-commutative 94.5%

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

      2. log1p-undefine 94.5%

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

      3. exp-to-pow 94.3%

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

      4. metadata-eval 94.3%

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

      5. pow-prod-up 94.3%

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

      6. +-commutative 94.3%

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

      7. pow1/3 95.7%

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

      8. +-commutative 95.7%

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

      9. pow1/3 98.3%

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

      10. pow2 98.3%

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

      11. +-commutative 98.3%

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

   8. Applied egg-rr 98.3%

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

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   10. Applied egg-rr 98.0%

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

       1. fma-undefine 98.0%

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

       2. rem-cube-cbrt 98.3%

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

       3. unpow2 98.3%

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

       4. cbrt-unprod 98.6%

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

       5. pow2 98.6%

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

   12. Applied egg-rr 98.6%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-commutative 91.7%

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

      6. +-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in x around 0 91.7%

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

   8. Taylor expanded in x around inf 91.7%

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

      1. *-commutative 91.7%

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

   10. Simplified 91.7%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} \cdot 2, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(cbrt(x), -0.1111111111111111, (0.3333333333333333 * pow(cbrt(x), 4.0))) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt((1.0 + x))), pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(cbrt(x), -0.1111111111111111, Float64(0.3333333333333333 * (cbrt(x) ^ 4.0))) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(Float64(1.0 + x))), (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111 + N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. div-inv 27.6%

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

      3. +-commutative 27.6%

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

      4. fma-define 27.6%

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

      5. *-commutative 27.6%

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

      6. pow-flip 27.7%

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

      7. metadata-eval 27.7%

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

   5. Applied egg-rr 27.7%

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

      1. *-lft-identity 27.7%

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

      2. fma-undefine 27.7%

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

      3. +-commutative 27.7%

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

      4. fma-define 27.7%

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

      5. unpow1/3 25.8%

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

      6. exp-to-pow 25.9%

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

      7. *-commutative 25.9%

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

      8. exp-prod 61.5%

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

      9. associate-*l* 61.5%

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

      10. *-commutative 61.5%

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

      11. exp-prod 61.5%

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

      12. exp-to-pow 61.2%

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

      13. unpow1/3 65.9%

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

   7. Simplified 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. *-un-lft-identity 65.9%

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

      3. fma-undefine 65.9%

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

      4. *-commutative 65.9%

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

      5. pow1/3 61.2%

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

      6. pow-pow 61.2%

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

      7. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}\right) \cdot {x}^{-2} \]

      8. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot {x}^{\color{blue}{\left(-2\right)}} \]

      9. pow-flip 59.8%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}} \]

      10. div-inv 59.8%

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

      11. unpow2 59.8%

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

      12. associate-/r* 88.1%

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

   9. Applied egg-rr 96.1%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      4. associate--l+ 91.1%

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

      5. +-commutative 91.1%

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

      6. +-commutative 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around 0 91.1%

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

      1. *-commutative 91.1%

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

      2. log1p-undefine 91.1%

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

      3. exp-to-pow 91.2%

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

   9. Applied egg-rr 91.2%

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

Alternative 7: 95.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{4}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} \cdot 2, e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      (cbrt x)
      -0.1111111111111111
      (* 0.3333333333333333 (pow (cbrt x) 4.0)))
     x)
    x)
   (/
    1.0
    (fma (cbrt x) (* (cbrt x) 2.0) (exp (* 0.6666666666666666 (log1p x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(cbrt(x), -0.1111111111111111, (0.3333333333333333 * pow(cbrt(x), 4.0))) / x) / x;
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) * 2.0), exp((0.6666666666666666 * log1p(x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(cbrt(x), -0.1111111111111111, Float64(0.3333333333333333 * (cbrt(x) ^ 4.0))) / x) / x);
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) * 2.0), exp(Float64(0.6666666666666666 * log1p(x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111 + N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] * 2.0), $MachinePrecision] + N[Exp[N[(0.6666666666666666 * N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. div-inv 27.6%

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

      3. +-commutative 27.6%

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

      4. fma-define 27.6%

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

      5. *-commutative 27.6%

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

      6. pow-flip 27.7%

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

      7. metadata-eval 27.7%

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

   5. Applied egg-rr 27.7%

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

      1. *-lft-identity 27.7%

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

      2. fma-undefine 27.7%

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

      3. +-commutative 27.7%

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

      4. fma-define 27.7%

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

      5. unpow1/3 25.8%

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

      6. exp-to-pow 25.9%

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

      7. *-commutative 25.9%

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

      8. exp-prod 61.5%

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

      9. associate-*l* 61.5%

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

      10. *-commutative 61.5%

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

      11. exp-prod 61.5%

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

      12. exp-to-pow 61.2%

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

      13. unpow1/3 65.9%

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

   7. Simplified 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. *-un-lft-identity 65.9%

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

      3. fma-undefine 65.9%

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

      4. *-commutative 65.9%

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

      5. pow1/3 61.2%

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

      6. pow-pow 61.2%

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

      7. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}\right) \cdot {x}^{-2} \]

      8. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot {x}^{\color{blue}{\left(-2\right)}} \]

      9. pow-flip 59.8%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}} \]

      10. div-inv 59.8%

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

      11. unpow2 59.8%

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

      12. associate-/r* 88.1%

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

   9. Applied egg-rr 96.1%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      4. associate--l+ 91.1%

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

      5. +-commutative 91.1%

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

      6. +-commutative 91.1%

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

   6. Simplified 91.1%

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

   7. Taylor expanded in x around 0 91.1%

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

   8. Taylor expanded in x around inf 91.1%

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

      1. *-commutative 91.1%

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

   10. Simplified 91.1%

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

Alternative 8: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt[3]{x}, -0.1111111111111111, 0.3333333333333333 \cdot {\left(\sqrt[3]{x}\right)}^{4}\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{2 \cdot {\left(\sqrt[3]{x}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55e+231)
   (/
    (/
     (fma
      (cbrt x)
      -0.1111111111111111
      (* 0.3333333333333333 (pow (cbrt x) 4.0)))
     x)
    x)
   (/ (+ 1.0 (- x x)) (* 2.0 (pow (cbrt x) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (fma(cbrt(x), -0.1111111111111111, (0.3333333333333333 * pow(cbrt(x), 4.0))) / x) / x;
	} else {
		tmp = (1.0 + (x - x)) / (2.0 * pow(cbrt(x), 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55e+231)
		tmp = Float64(Float64(fma(cbrt(x), -0.1111111111111111, Float64(0.3333333333333333 * (cbrt(x) ^ 4.0))) / x) / x);
	else
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(2.0 * (cbrt(x) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55e+231], N[(N[(N[(N[Power[x, 1/3], $MachinePrecision] * -0.1111111111111111 + N[(0.3333333333333333 * N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.54999999999999995e231

   1. Initial program 7.2%

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

   3. Taylor expanded in x around inf 27.6%

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

      1. *-un-lft-identity 27.6%

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

      2. div-inv 27.6%

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

      3. +-commutative 27.6%

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

      4. fma-define 27.6%

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

      5. *-commutative 27.6%

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

      6. pow-flip 27.7%

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

      7. metadata-eval 27.7%

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

   5. Applied egg-rr 27.7%

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

      1. *-lft-identity 27.7%

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

      2. fma-undefine 27.7%

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

      3. +-commutative 27.7%

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

      4. fma-define 27.7%

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

      5. unpow1/3 25.8%

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

      6. exp-to-pow 25.9%

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

      7. *-commutative 25.9%

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

      8. exp-prod 61.5%

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

      9. associate-*l* 61.5%

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

      10. *-commutative 61.5%

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

      11. exp-prod 61.5%

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

      12. exp-to-pow 61.2%

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

      13. unpow1/3 65.9%

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

   7. Simplified 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. *-un-lft-identity 65.9%

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

      3. fma-undefine 65.9%

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

      4. *-commutative 65.9%

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

      5. pow1/3 61.2%

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

      6. pow-pow 61.2%

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

      7. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{\color{blue}{1.3333333333333333}}\right) \cdot {x}^{-2} \]

      8. metadata-eval 61.2%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot {x}^{\color{blue}{\left(-2\right)}} \]

      9. pow-flip 59.8%

         \[\leadsto \left(-0.1111111111111111 \cdot \sqrt[3]{x} + 0.3333333333333333 \cdot {x}^{1.3333333333333333}\right) \cdot \color{blue}{\frac{1}{{x}^{2}}} \]

      10. div-inv 59.8%

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

      11. unpow2 59.8%

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

      12. associate-/r* 88.1%

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

   9. Applied egg-rr 96.1%

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

   if 1.54999999999999995e231 < x

   1. Initial program 5.1%

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

      1. flip3-- 5.1%

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

      2. div-inv 5.1%

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

      3. rem-cube-cbrt 3.1%

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

      4. rem-cube-cbrt 5.1%

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

      5. +-commutative 5.1%

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

      6. distribute-rgt-out 5.1%

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

      7. +-commutative 5.1%

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

      8. fma-define 5.1%

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

      9. add-exp-log 5.1%

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

   4. Applied egg-rr 5.1%

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

      1. associate-*r/ 5.1%

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

      2. *-rgt-identity 5.1%

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

      3. +-commutative 5.1%

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

      4. associate--l+ 91.1%

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

      5. +-commutative 91.1%

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

      6. +-commutative 91.1%

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

   6. Simplified 91.1%

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

      1. *-commutative 91.1%

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

      2. log1p-undefine 91.1%

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

      3. exp-to-pow 91.2%

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

      4. metadata-eval 91.2%

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

      5. pow-prod-up 91.2%

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

      6. +-commutative 91.2%

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

      7. pow1/3 92.7%

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

      8. +-commutative 92.7%

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

      9. pow1/3 98.4%

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

      10. pow2 98.4%

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

      11. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in x around inf 5.1%

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

       1. unpow2 5.1%

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

       2. rem-cube-cbrt 5.1%

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

       3. rem-cube-cbrt 5.1%

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

       4. cube-prod 5.1%

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

       5. unpow2 5.1%

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

       6. rem-cbrt-cube 20.0%

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

   11. Simplified 20.0%

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

Alternative 9: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.7%

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

   3. Taylor expanded in x around inf 94.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-commutative 91.7%

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

      6. +-commutative 91.7%

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

   6. Simplified 91.7%

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

      1. *-commutative 91.7%

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

      2. log1p-undefine 91.7%

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

      3. exp-to-pow 91.5%

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

      4. metadata-eval 91.5%

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

      5. pow-prod-up 91.5%

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

      6. +-commutative 91.5%

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

      7. pow1/3 93.0%

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

      8. +-commutative 93.0%

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

      9. pow1/3 98.3%

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

      10. pow2 98.3%

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

      11. +-commutative 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in x around inf 4.7%

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

       1. unpow2 4.7%

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

       2. rem-cube-cbrt 4.7%

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

       3. rem-cube-cbrt 4.7%

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

       4. cube-prod 4.7%

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

       5. unpow2 4.7%

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

       6. rem-cbrt-cube 20.0%

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

   11. Simplified 20.0%

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

Alternative 10: 56.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 8.7%

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

   3. Taylor expanded in x around inf 94.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 4.7%

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

      1. flip3-- 4.7%

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

      2. div-inv 4.7%

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

      3. rem-cube-cbrt 3.2%

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

      4. rem-cube-cbrt 4.7%

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

      5. +-commutative 4.7%

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

      6. distribute-rgt-out 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-define 4.7%

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

      9. add-exp-log 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 91.7%

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

      5. +-commutative 91.7%

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

      6. +-commutative 91.7%

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

   6. Simplified 91.7%

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

      1. *-commutative 91.7%

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

      2. log1p-undefine 91.7%

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

      3. exp-to-pow 91.5%

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

      4. metadata-eval 91.5%

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

      5. pow-prod-up 91.5%

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

      6. +-commutative 91.5%

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

      7. pow1/3 93.0%

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

      8. +-commutative 93.0%

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

      9. pow1/3 98.3%

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

      10. pow2 98.3%

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

      11. +-commutative 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in x around 0 17.7%

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

Alternative 11: 49.9% 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 6.6%

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

3. Taylor expanded in x around inf 47.4%

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

Alternative 12: 7.1% 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 6.6%

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

3. Final simplification 6.6%

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

Alternative 13: 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 6.6%

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

3. Taylor expanded in x around 0 1.8%

   \[\leadsto \color{blue}{1 - \sqrt[3]{x}} \]
4. Step-by-step derivation

   1. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.2%

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

   5. fabs-neg 5.2%

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

   6. unpow1/3 5.2%

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

   7. metadata-eval 5.2%

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

   8. pow-sqr 5.2%

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

   9. fabs-sqr 5.2%

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

   10. pow-sqr 5.2%

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

   11. metadata-eval 5.2%

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

   12. unpow1/3 5.2%

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

5. Simplified 5.2%

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

6. Taylor expanded in x around inf 5.2%

   \[\leadsto \color{blue}{\sqrt[3]{x}} \]
7. 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 2024172 
(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)))