2cbrt (problem 3.3.4)

Percentage Accurate: 9.2% → 99.1%

Time: 28.3s

Alternatives: 9

Speedup: 2.0×

• could not determine a ground truth

  1. x = 4.3145757869733155e+167

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: 9.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. add-exp-log 7.5%

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

4. Applied egg-rr 7.5%

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

   1. rem-exp-log 7.5%

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

   2. flip3-- 7.5%

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

   3. +-commutative 7.5%

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

   4. +-commutative 7.5%

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

   5. rem-cube-cbrt 7.7%

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

   6. rem-cube-cbrt 11.1%

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

   7. div-sub 7.9%

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

6. Applied egg-rr 7.8%

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

   1. div-sub 11.1%

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

   2. associate--l+ 98.4%

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

   3. +-inverses 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. fma-define 98.4%

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

   7. +-commutative 98.4%

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

   8. +-commutative 98.4%

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

8. Simplified 98.4%

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

   1. fma-undefine 98.4%

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

   2. +-commutative 98.4%

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

   3. distribute-rgt-in 98.4%

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

   4. associate-+r+ 98.4%

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

   5. pow2 98.4%

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

   6. cbrt-unprod 98.8%

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

10. Applied egg-rr 98.8%

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

    1. associate-+l+ 98.7%

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

    2. +-commutative 98.7%

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

    3. *-commutative 98.7%

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

    4. +-commutative 98.7%

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

12. Simplified 98.7%

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

13. Taylor expanded in x around 0 99.0%

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

14. Final simplification 99.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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) * (t_0 + Math.cbrt(x))));
}

Julia

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

Derivation

1. Initial program 7.5%

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

   1. add-exp-log 7.5%

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

4. Applied egg-rr 7.5%

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

   1. rem-exp-log 7.5%

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

   2. flip3-- 7.5%

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

   3. +-commutative 7.5%

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

   4. +-commutative 7.5%

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

   5. rem-cube-cbrt 7.7%

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

   6. rem-cube-cbrt 11.1%

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

   7. div-sub 7.9%

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

6. Applied egg-rr 7.8%

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

   1. div-sub 11.1%

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

   2. associate--l+ 98.4%

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

   3. +-inverses 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. fma-define 98.4%

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

   7. +-commutative 98.4%

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

   8. +-commutative 98.4%

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

8. Simplified 98.4%

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

   1. fma-undefine 98.4%

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

   2. +-commutative 98.4%

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

   3. +-commutative 98.4%

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

10. Applied egg-rr 98.4%

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

11. Final simplification 98.4%

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

Alternative 3: 97.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

   1. add-sqr-sqrt 7.8%

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

   2. add-sqr-sqrt 7.3%

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

   3. difference-of-squares 7.4%

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

   4. pow1/3 7.4%

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

   5. sqrt-pow1 7.4%

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

   6. metadata-eval 7.4%

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

   7. pow1/3 7.4%

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

   8. sqrt-pow1 7.4%

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

   9. metadata-eval 7.4%

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

   10. pow1/3 5.3%

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

   11. sqrt-pow1 5.3%

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

   12. metadata-eval 5.3%

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

   13. pow1/3 7.5%

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

   14. sqrt-pow1 7.4%

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

   15. metadata-eval 7.4%

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

4. Applied egg-rr 7.4%

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

5. Taylor expanded in x around inf 97.7%

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

   1. associate-+r+ 97.7%

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

   2. distribute-rgt-out 97.7%

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

   3. metadata-eval 97.7%

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 4: 95.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 96.2%

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

   1. cbrt-div 96.5%

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

   2. metadata-eval 96.5%

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

5. Applied egg-rr 96.5%

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

6. Final simplification 96.5%

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

Alternative 5: 94.8% accurate, 1.0× speedup

Math

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

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 7.5%

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

   1. add-exp-log 7.5%

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

4. Applied egg-rr 7.5%

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

5. Taylor expanded in x around inf 96.2%

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

   1. *-commutative 96.2%

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

   2. exp-to-pow 90.9%

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

   3. *-commutative 90.9%

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

   4. rec-exp 90.9%

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

   5. mul-1-neg 90.9%

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

   6. associate-*r* 90.9%

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

   7. metadata-eval 90.9%

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

   8. *-commutative 90.9%

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

   9. exp-to-pow 96.2%

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

7. Simplified 96.2%

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

8. Final simplification 96.2%

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

Alternative 6: 95.1% 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 7.5%

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

3. Taylor expanded in x around inf 96.2%

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

   1. cbrt-div 96.5%

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

   2. metadata-eval 96.5%

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

   3. unpow2 96.5%

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

   4. cbrt-prod 95.8%

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

   5. un-div-inv 95.8%

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

   6. cbrt-prod 96.5%

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

   7. unpow2 96.5%

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

5. Applied egg-rr 96.5%

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

6. Final simplification 96.5%

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

Alternative 7: 94.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 96.2%

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

   1. add-sqr-sqrt 96.2%

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

   2. sqrt-div 96.2%

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

   3. metadata-eval 96.2%

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

   4. sqrt-pow1 96.2%

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

   5. metadata-eval 96.2%

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

   6. pow1 96.2%

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

   7. sqrt-div 96.2%

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

   8. metadata-eval 96.2%

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

   9. sqrt-pow1 96.2%

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

   10. metadata-eval 96.2%

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

   11. pow1 96.2%

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

5. Applied egg-rr 96.2%

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

6. Final simplification 96.2%

   \[\leadsto 0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x} \cdot \frac{1}{x}} \]
7. Add Preprocessing

Alternative 8: 88.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around inf 96.2%

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

   1. pow1/3 89.6%

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

   2. pow-flip 89.6%

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

   3. pow-pow 89.6%

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

   4. metadata-eval 89.6%

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

   5. metadata-eval 89.6%

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

5. Applied egg-rr 89.6%

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

6. Final simplification 89.6%

   \[\leadsto 0.3333333333333333 \cdot {x}^{-0.6666666666666666} \]
7. Add Preprocessing

Alternative 9: 6.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

function code(x)
	return cbrt(x)
end

Wolfram

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

Derivation

1. Initial program 7.5%

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

3. Taylor expanded in x around 0 1.7%

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

   1. sub-neg 1.7%

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

   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 6.8%

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

   5. fabs-neg 6.8%

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

   6. unpow1/3 6.8%

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

   7. metadata-eval 6.8%

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

   8. pow-sqr 6.8%

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

   9. fabs-sqr 6.8%

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

   10. pow-sqr 6.8%

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

   11. metadata-eval 6.8%

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

   12. unpow1/3 6.8%

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

5. Simplified 6.8%

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

6. Taylor expanded in x around inf 6.8%

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

7. Final simplification 6.8%

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

Developer target: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024083 
(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)))