2cbrt (problem 3.3.4)

Percentage Accurate: 6.8% → 98.5%

Time: 23.6s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.2× speedup

Math

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

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 6.5%

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

   1. pow1/3 7.7%

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

   2. add-sqr-sqrt 7.7%

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

   3. pow2 7.7%

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

   4. pow-pow 7.7%

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

   5. metadata-eval 7.7%

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

4. Applied egg-rr 7.7%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666}} \]
5. Step-by-step derivation

   1. sqrt-pow2 7.7%

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

   2. metadata-eval 7.7%

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

   3. pow1/3 6.5%

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

   4. flip3-- 6.8%

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

   5. div-inv 6.8%

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

   6. pow3 6.6%

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

   7. add-cube-cbrt 6.4%

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

   8. rem-cube-cbrt 9.3%

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

   9. distribute-rgt-in 9.3%

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

   10. +-commutative 9.3%

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

   11. +-commutative 9.3%

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

6. Applied egg-rr 9.3%

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

   1. associate-*r/ 9.3%

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

   2. *-rgt-identity 9.3%

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

   3. +-commutative 9.3%

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

   4. associate--l+ 98.5%

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

   5. +-inverses 98.5%

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

   6. metadata-eval 98.5%

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

   7. fma-define 98.5%

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

   8. +-commutative 98.5%

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

   9. +-commutative 98.5%

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

8. Simplified 98.5%

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

   1. pow1/3 94.5%

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

   2. add-sqr-sqrt 94.5%

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

   3. unpow-prod-down 94.5%

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

10. Applied egg-rr 94.5%

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

    1. unpow1/3 95.8%

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

    2. unpow1/3 98.7%

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

12. Simplified 98.7%

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

Alternative 2: 98.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. pow1/3 7.7%

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

   2. add-sqr-sqrt 7.7%

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

   3. pow2 7.7%

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

   4. pow-pow 7.7%

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

   5. metadata-eval 7.7%

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

4. Applied egg-rr 7.7%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666}} \]
5. Step-by-step derivation

   1. sqrt-pow2 7.7%

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

   2. metadata-eval 7.7%

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

   3. pow1/3 6.5%

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

   4. flip3-- 6.8%

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

   5. div-inv 6.8%

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

   6. pow3 6.6%

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

   7. add-cube-cbrt 6.4%

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

   8. rem-cube-cbrt 9.3%

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

   9. distribute-rgt-in 9.3%

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

   10. +-commutative 9.3%

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

   11. +-commutative 9.3%

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

6. Applied egg-rr 9.3%

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

   1. associate-*r/ 9.3%

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

   2. *-rgt-identity 9.3%

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

   3. +-commutative 9.3%

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

   4. associate--l+ 98.5%

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

   5. +-inverses 98.5%

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

   6. metadata-eval 98.5%

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

   7. fma-define 98.5%

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

   8. +-commutative 98.5%

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

   9. +-commutative 98.5%

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

8. Simplified 98.5%

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

   1. pow1/3 94.5%

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

   2. add-sqr-sqrt 94.5%

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

   3. unpow-prod-down 94.5%

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

10. Applied egg-rr 93.1%

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

    1. unpow1/3 95.8%

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

    2. unpow1/3 98.7%

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

12. Simplified 98.6%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. pow1/3 7.7%

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

   2. add-sqr-sqrt 7.7%

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

   3. pow2 7.7%

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

   4. pow-pow 7.7%

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

   5. metadata-eval 7.7%

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

4. Applied egg-rr 7.7%

   \[\leadsto \sqrt[3]{x + 1} - \color{blue}{{\left(\sqrt{x}\right)}^{0.6666666666666666}} \]
5. Step-by-step derivation

   1. sqrt-pow2 7.7%

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

   2. metadata-eval 7.7%

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

   3. pow1/3 6.5%

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

   4. flip3-- 6.8%

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

   5. div-inv 6.8%

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

   6. pow3 6.6%

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

   7. add-cube-cbrt 6.4%

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

   8. rem-cube-cbrt 9.3%

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

   9. distribute-rgt-in 9.3%

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

   10. +-commutative 9.3%

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

   11. +-commutative 9.3%

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

6. Applied egg-rr 9.3%

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

   1. associate-*r/ 9.3%

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

   2. *-rgt-identity 9.3%

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

   3. +-commutative 9.3%

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

   4. associate--l+ 98.5%

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

   5. +-inverses 98.5%

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

   6. metadata-eval 98.5%

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

   7. fma-define 98.5%

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

   8. +-commutative 98.5%

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

   9. +-commutative 98.5%

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

8. Simplified 98.5%

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

   1. fma-undefine 98.5%

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

   2. +-commutative 98.5%

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

   3. +-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 4: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt[3]{x} \cdot 0.3333333333333333 + \sqrt[3]{\frac{1}{{x}^{2}}} \cdot -0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (+
   (* (cbrt x) 0.3333333333333333)
   (* (cbrt (/ 1.0 (pow x 2.0))) -0.1111111111111111))
  x))

C

double code(double x) {
	return ((cbrt(x) * 0.3333333333333333) + (cbrt((1.0 / pow(x, 2.0))) * -0.1111111111111111)) / x;
}

Java

public static double code(double x) {
	return ((Math.cbrt(x) * 0.3333333333333333) + (Math.cbrt((1.0 / Math.pow(x, 2.0))) * -0.1111111111111111)) / x;
}

Julia

function code(x)
	return Float64(Float64(Float64(cbrt(x) * 0.3333333333333333) + Float64(cbrt(Float64(1.0 / (x ^ 2.0))) * -0.1111111111111111)) / x)
end

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. add-sqr-sqrt 6.1%

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

   2. add-sqr-sqrt 6.5%

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

   3. difference-of-squares 6.5%

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

   4. pow1/3 6.5%

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

   5. sqrt-pow1 6.5%

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

   6. metadata-eval 6.5%

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

   7. pow1/3 6.5%

      \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

   8. sqrt-pow1 6.5%

      \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}} - \sqrt{\sqrt[3]{x}}\right) \]

   9. metadata-eval 6.5%

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

   10. pow1/3 4.1%

       \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\sqrt{\color{blue}{{\left(x + 1\right)}^{0.3333333333333333}}} - \sqrt{\sqrt[3]{x}}\right) \]

   11. sqrt-pow1 4.1%

       \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} - \sqrt{\sqrt[3]{x}}\right) \]

   12. metadata-eval 4.1%

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

   13. pow1/3 6.5%

       \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - \sqrt{\color{blue}{{x}^{0.3333333333333333}}}\right) \]

   14. sqrt-pow1 6.6%

       \[\leadsto \left({\left(x + 1\right)}^{0.16666666666666666} + {x}^{0.16666666666666666}\right) \cdot \left({\left(x + 1\right)}^{0.16666666666666666} - \color{blue}{{x}^{\left(\frac{0.3333333333333333}{2}\right)}}\right) \]

   15. metadata-eval 6.6%

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

4. Applied egg-rr 6.6%

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

5. Taylor expanded in x around inf 98.3%

   \[\leadsto \color{blue}{\frac{-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + \left(0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.3333333333333333 \cdot \sqrt[3]{x}\right)}{x}} \]
6. Step-by-step derivation

   1. associate-+r+ 98.4%

      \[\leadsto \frac{\color{blue}{\left(-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right) + 0.3333333333333333 \cdot \sqrt[3]{x}}}{x} \]

   2. +-commutative 98.4%

      \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot \sqrt[3]{x} + \left(-0.1388888888888889 \cdot \sqrt[3]{\frac{1}{{x}^{2}}} + 0.027777777777777776 \cdot \sqrt[3]{\frac{1}{{x}^{2}}}\right)}}{x} \]

   3. distribute-rgt-out 98.4%

      \[\leadsto \frac{0.3333333333333333 \cdot \sqrt[3]{x} + \color{blue}{\sqrt[3]{\frac{1}{{x}^{2}}} \cdot \left(-0.1388888888888889 + 0.027777777777777776\right)}}{x} \]

   4. metadata-eval 98.4%

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

7. Simplified 98.4%

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

8. Final simplification 98.4%

   \[\leadsto \frac{\sqrt[3]{x} \cdot 0.3333333333333333 + \sqrt[3]{\frac{1}{{x}^{2}}} \cdot -0.1111111111111111}{x} \]
9. Add Preprocessing

Alternative 5: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.55e+231) {
		tmp = (pow(cbrt(x), 4.0) / x) * (0.3333333333333333 / x);
	} 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 <= 1.55e+231) {
		tmp = (Math.pow(Math.cbrt(x), 4.0) / x) * (0.3333333333333333 / x);
	} 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 <= 1.55e+231)
		tmp = Float64(Float64((cbrt(x) ^ 4.0) / x) * Float64(0.3333333333333333 / x));
	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, 1.55e+231], N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] / x), $MachinePrecision] * N[(0.3333333333333333 / x), $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 < 1.54999999999999995e231

   1. Initial program 7.0%

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

   3. Taylor expanded in x around inf 32.8%

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

      1. +-commutative 32.8%

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

      2. fma-define 32.8%

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

   5. Simplified 32.8%

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

   6. Taylor expanded in x around inf 31.6%

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

      1. *-commutative 31.6%

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

      2. unpow2 31.6%

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

      3. times-frac 32.3%

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

      4. pow1/3 30.2%

         \[\leadsto \frac{\color{blue}{{\left({x}^{4}\right)}^{0.3333333333333333}}}{x} \cdot \frac{0.3333333333333333}{x} \]

      5. pow-pow 87.9%

         \[\leadsto \frac{\color{blue}{{x}^{\left(4 \cdot 0.3333333333333333\right)}}}{x} \cdot \frac{0.3333333333333333}{x} \]

      6. metadata-eval 87.9%

         \[\leadsto \frac{{x}^{\color{blue}{1.3333333333333333}}}{x} \cdot \frac{0.3333333333333333}{x} \]

      7. metadata-eval 87.9%

         \[\leadsto \frac{{x}^{\color{blue}{\left(0.3333333333333333 \cdot 4\right)}}}{x} \cdot \frac{0.3333333333333333}{x} \]

      8. pow-pow 87.9%

         \[\leadsto \frac{\color{blue}{{\left({x}^{0.3333333333333333}\right)}^{4}}}{x} \cdot \frac{0.3333333333333333}{x} \]

      9. pow1/3 95.8%

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

   8. Applied egg-rr 95.8%

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

   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 3.3%

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

      4. rem-cube-cbrt 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+ 92.2%

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

      5. +-inverses 92.2%

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

      6. metadata-eval 92.2%

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

      7. +-commutative 92.2%

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

      8. exp-prod 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in x around 0 17.7%

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

Alternative 6: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\sqrt[3]{x}\right)}^{4}}{x} \cdot \frac{0.3333333333333333}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (pow (cbrt x) 4.0) x) (/ 0.3333333333333333 x)))

C

double code(double x) {
	return (pow(cbrt(x), 4.0) / x) * (0.3333333333333333 / x);
}

Java

public static double code(double x) {
	return (Math.pow(Math.cbrt(x), 4.0) / x) * (0.3333333333333333 / x);
}

Julia

function code(x)
	return Float64(Float64((cbrt(x) ^ 4.0) / x) * Float64(0.3333333333333333 / x))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 4.0], $MachinePrecision] / x), $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 25.3%

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

   1. +-commutative 25.3%

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

   2. fma-define 25.3%

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

5. Simplified 25.3%

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

6. Taylor expanded in x around inf 24.4%

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

   1. *-commutative 24.4%

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

   2. unpow2 24.4%

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

   3. times-frac 25.3%

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

   4. pow1/3 23.7%

      \[\leadsto \frac{\color{blue}{{\left({x}^{4}\right)}^{0.3333333333333333}}}{x} \cdot \frac{0.3333333333333333}{x} \]

   5. pow-pow 68.1%

      \[\leadsto \frac{\color{blue}{{x}^{\left(4 \cdot 0.3333333333333333\right)}}}{x} \cdot \frac{0.3333333333333333}{x} \]

   6. metadata-eval 68.1%

      \[\leadsto \frac{{x}^{\color{blue}{1.3333333333333333}}}{x} \cdot \frac{0.3333333333333333}{x} \]

   7. metadata-eval 68.1%

      \[\leadsto \frac{{x}^{\color{blue}{\left(0.3333333333333333 \cdot 4\right)}}}{x} \cdot \frac{0.3333333333333333}{x} \]

   8. pow-pow 68.1%

      \[\leadsto \frac{\color{blue}{{\left({x}^{0.3333333333333333}\right)}^{4}}}{x} \cdot \frac{0.3333333333333333}{x} \]

   9. pow1/3 74.2%

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

8. Applied egg-rr 74.2%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{4}}{x} \cdot \frac{0.3333333333333333}{x}} \]
9. Add Preprocessing

Alternative 7: 5.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 4.1%

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

   1. pow1/3 5.9%

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

5. Applied egg-rr 5.9%

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

6. Final simplification 5.9%

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

Alternative 8: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around inf 25.3%

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

   1. +-commutative 25.3%

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

   2. fma-define 25.3%

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

5. Simplified 25.3%

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

6. Taylor expanded in x around inf 24.4%

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

   1. *-un-lft-identity 24.4%

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

   2. add-cbrt-cube 13.8%

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

   3. pow-sqr 13.8%

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

   4. metadata-eval 13.8%

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

   5. cbrt-prod 23.5%

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

   6. unpow2 23.5%

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

   7. cbrt-prod 23.4%

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

   8. times-frac 23.4%

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

   9. pow1/3 21.9%

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

   10. pow-pow 23.2%

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

   11. metadata-eval 23.2%

       \[\leadsto \frac{1}{{x}^{\color{blue}{1.3333333333333333}}} \cdot \frac{0.3333333333333333 \cdot \sqrt[3]{{x}^{4}}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \]

   12. metadata-eval 23.2%

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

   13. pow-pow 23.2%

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

   14. pow1/3 24.5%

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

8. Applied egg-rr 54.2%

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

   1. associate-*r/ 54.2%

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

   2. *-commutative 54.2%

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

   3. associate-*r* 54.1%

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

   4. lft-mult-inverse 55.4%

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

   5. metadata-eval 55.4%

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

10. Simplified 55.4%

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

Alternative 9: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

3. Taylor expanded in x around 0 1.8%

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

   1. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.4%

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

   5. fabs-neg 5.4%

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

   6. unpow1/3 5.4%

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

   7. metadata-eval 5.4%

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

   8. pow-sqr 5.4%

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

   9. fabs-sqr 5.4%

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

   10. pow-sqr 5.4%

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

   11. metadata-eval 5.4%

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

   12. unpow1/3 5.4%

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

5. Simplified 5.4%

   \[\leadsto \color{blue}{1 + \sqrt[3]{x}} \]
6. Add Preprocessing

Developer target: 98.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (/ 1.0 (+ (+ (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0))) (* (cbrt x) (cbrt (+ x 1.0)))) (* (cbrt x) (cbrt x))))

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