2cbrt (problem 3.3.4)

Percentage Accurate: 53.3% → 99.2%

Time: 16.0s

Alternatives: 11

Speedup: 1.0×

Specification

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: 53.3% 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: 99.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. flip3-- 52.8%

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

   2. div-inv 52.8%

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

   3. rem-cube-cbrt 52.7%

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

   4. rem-cube-cbrt 53.2%

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

   5. cbrt-unprod 53.2%

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

   6. pow2 53.2%

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

   7. distribute-rgt-out 53.2%

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

   8. +-commutative 53.2%

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

3. Applied egg-rr 53.2%

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

   1. associate-*r/ 53.2%

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

   2. *-rgt-identity 53.2%

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

   3. +-commutative 53.2%

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

   4. associate--l+ 75.5%

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

   5. +-inverses 75.5%

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

   6. metadata-eval 75.5%

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

   7. +-commutative 75.5%

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

   8. fma-def 75.5%

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

   9. +-commutative 75.5%

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

   10. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. unpow2 75.5%

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

   2. cbrt-prod 99.2%

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

7. Applied egg-rr 99.2%

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

   1. unpow2 99.2%

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

9. Simplified 99.2%

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

    1. unpow2 99.2%

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

    2. flip-+ 75.4%

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

    3. metadata-eval 75.4%

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

    4. cbrt-undiv 75.5%

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

    5. clear-num 75.5%

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

    6. flip-+ 75.5%

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

    7. metadata-eval 75.5%

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

    8. cbrt-undiv 75.5%

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

    9. clear-num 75.5%

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

    10. frac-times 75.5%

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

    11. metadata-eval 75.5%

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

11. Applied egg-rr 99.2%

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

    1. pow2 99.2%

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

    2. add-cbrt-cube 99.1%

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

    3. frac-times 99.1%

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

    4. metadata-eval 99.1%

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

    5. frac-times 99.1%

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

    6. metadata-eval 99.1%

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

    7. add-cube-cbrt 99.2%

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

13. Applied egg-rr 99.2%

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

14. Final simplification 99.2%

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

Alternative 2: 99.2% 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 52.8%

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

   1. flip3-- 52.8%

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

   2. div-inv 52.8%

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

   3. rem-cube-cbrt 52.7%

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

   4. rem-cube-cbrt 53.2%

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

   5. cbrt-unprod 53.2%

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

   6. pow2 53.2%

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

   7. distribute-rgt-out 53.2%

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

   8. +-commutative 53.2%

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

3. Applied egg-rr 53.2%

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

   1. associate-*r/ 53.2%

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

   2. *-rgt-identity 53.2%

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

   3. +-commutative 53.2%

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

   4. associate--l+ 75.5%

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

   5. +-inverses 75.5%

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

   6. metadata-eval 75.5%

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

   7. +-commutative 75.5%

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

   8. fma-def 75.5%

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

   9. +-commutative 75.5%

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

   10. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. unpow2 75.5%

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

   2. cbrt-prod 99.2%

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

7. Applied egg-rr 99.2%

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

   1. unpow2 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 88.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_1}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x + x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_1, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ (cbrt x) t_0)))
   (if (<= x -1.35e+154)
     (/ 1.0 (+ 1.0 (* (cbrt x) t_1)))
     (if (<= x 1.32e+154)
       (/ 1.0 (+ (pow t_0 2.0) (+ (cbrt (* x x)) (cbrt (+ x (* x x))))))
       (/ 1.0 (fma (cbrt x) t_1 (pow (+ 1.0 x) 0.6666666666666666)))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = cbrt(x) + t_0;
	double tmp;
	if (x <= -1.35e+154) {
		tmp = 1.0 / (1.0 + (cbrt(x) * t_1));
	} else if (x <= 1.32e+154) {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt((x * x)) + cbrt((x + (x * x)))));
	} else {
		tmp = 1.0 / fma(cbrt(x), t_1, pow((1.0 + x), 0.6666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(cbrt(x) + t_0)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * t_1)));
	elseif (x <= 1.32e+154)
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(Float64(x * x)) + cbrt(Float64(x + Float64(x * x))))));
	else
		tmp = Float64(1.0 / fma(cbrt(x), t_1, (Float64(1.0 + x) ^ 0.6666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e+154], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * t$95$1 + N[Power[N[(1.0 + x), $MachinePrecision], 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. 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 3.6%

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

      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. cbrt-unprod 4.7%

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

      6. pow2 4.7%

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

      7. distribute-rgt-out 4.7%

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

      8. +-commutative 4.7%

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

   3. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. +-inverses 4.7%

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

      6. metadata-eval 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-def 4.7%

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

      9. +-commutative 4.7%

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

      10. +-commutative 4.7%

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

   5. Simplified 4.7%

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

      1. unpow2 4.7%

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

      2. cbrt-prod 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. unpow2 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in x around 0 20.0%

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

       1. fma-udef 20.0%

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

   12. Applied egg-rr 20.0%

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

   if -1.35000000000000003e154 < x < 1.31999999999999998e154

   1. Initial program 69.2%

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

      1. add-exp-log 68.9%

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

      2. pow1/3 68.5%

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

      3. log-pow 68.6%

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

      4. +-commutative 68.6%

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

      5. log1p-udef 68.6%

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

   3. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. *-commutative 68.6%

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

      2. exp-prod 68.5%

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

      3. unpow1/3 68.9%

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

   5. Simplified 68.9%

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

      1. flip3-- 68.9%

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

      2. div-inv 68.9%

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

      3. log1p-udef 68.9%

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

      4. add-exp-log 68.6%

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

      5. rem-cube-cbrt 68.7%

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

      6. rem-cube-cbrt 69.0%

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

   7. Applied egg-rr 69.7%

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

      1. associate-*r/ 69.7%

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

      2. *-rgt-identity 69.7%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. *-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

   if 1.31999999999999998e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. 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 2.9%

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

      4. rem-cube-cbrt 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. cbrt-unprod 4.7%

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

      6. pow2 4.7%

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

      7. distribute-rgt-out 4.7%

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

      8. +-commutative 4.7%

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

   3. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. +-inverses 4.7%

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

      6. metadata-eval 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-def 4.7%

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

      9. +-commutative 4.7%

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

      10. +-commutative 4.7%

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

   5. Simplified 4.7%

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

      1. pow1/3 4.7%

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

      2. pow-pow 91.6%

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

      3. metadata-eval 91.6%

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

   7. Applied egg-rr 91.6%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x + x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 4: 99.2% 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 52.8%

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

   1. flip3-- 52.8%

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

   2. div-inv 52.8%

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

   3. rem-cube-cbrt 52.7%

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

   4. rem-cube-cbrt 53.2%

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

   5. cbrt-unprod 53.2%

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

   6. pow2 53.2%

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

   7. distribute-rgt-out 53.2%

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

   8. +-commutative 53.2%

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

3. Applied egg-rr 53.2%

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

   1. associate-*r/ 53.2%

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

   2. *-rgt-identity 53.2%

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

   3. +-commutative 53.2%

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

   4. associate--l+ 75.5%

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

   5. +-inverses 75.5%

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

   6. metadata-eval 75.5%

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

   7. +-commutative 75.5%

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

   8. fma-def 75.5%

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

   9. +-commutative 75.5%

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

   10. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. unpow2 75.5%

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

   2. cbrt-prod 99.2%

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

7. Applied egg-rr 99.2%

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

   1. unpow2 99.2%

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

9. Simplified 99.2%

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

    1. fma-udef 99.2%

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

11. Applied egg-rr 99.2%

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

12. Final simplification 99.2%

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

Alternative 5: 79.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154} \lor \neg \left(x \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x + x \cdot x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (or (<= x -1.35e+154) (not (<= x 1.32e+154)))
     (/ 1.0 (+ 1.0 (* (cbrt x) (+ (cbrt x) t_0))))
     (/ 1.0 (+ (pow t_0 2.0) (+ (cbrt (* x x)) (cbrt (+ x (* x x)))))))))

C

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double tmp;
	if ((x <= -1.35e+154) || !(x <= 1.32e+154)) {
		tmp = 1.0 / (1.0 + (cbrt(x) * (cbrt(x) + t_0)));
	} else {
		tmp = 1.0 / (pow(t_0, 2.0) + (cbrt((x * x)) + cbrt((x + (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.cbrt((1.0 + x));
	double tmp;
	if ((x <= -1.35e+154) || !(x <= 1.32e+154)) {
		tmp = 1.0 / (1.0 + (Math.cbrt(x) * (Math.cbrt(x) + t_0)));
	} else {
		tmp = 1.0 / (Math.pow(t_0, 2.0) + (Math.cbrt((x * x)) + Math.cbrt((x + (x * x)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if ((x <= -1.35e+154) || !(x <= 1.32e+154))
		tmp = Float64(1.0 / Float64(1.0 + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	else
		tmp = Float64(1.0 / Float64((t_0 ^ 2.0) + Float64(cbrt(Float64(x * x)) + cbrt(Float64(x + Float64(x * x))))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[Or[LessEqual[x, -1.35e+154], N[Not[LessEqual[x, 1.32e+154]], $MachinePrecision]], N[(1.0 / N[(1.0 + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(N[Power[N[(x * x), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154 or 1.31999999999999998e154 < x

   1. Initial program 4.7%

      \[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
   2. 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 3.3%

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

      4. rem-cube-cbrt 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. cbrt-unprod 4.7%

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

      6. pow2 4.7%

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

      7. distribute-rgt-out 4.7%

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

      8. +-commutative 4.7%

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

   3. Applied egg-rr 4.7%

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

      1. associate-*r/ 4.7%

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

      2. *-rgt-identity 4.7%

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

      3. +-commutative 4.7%

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

      4. associate--l+ 4.7%

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

      5. +-inverses 4.7%

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

      6. metadata-eval 4.7%

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

      7. +-commutative 4.7%

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

      8. fma-def 4.7%

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

      9. +-commutative 4.7%

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

      10. +-commutative 4.7%

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

   5. Simplified 4.7%

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

      1. unpow2 4.7%

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

      2. cbrt-prod 98.3%

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

   7. Applied egg-rr 98.3%

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

      1. unpow2 98.3%

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

   9. Simplified 98.3%

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

   10. Taylor expanded in x around 0 20.0%

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

       1. fma-udef 20.0%

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

   12. Applied egg-rr 20.0%

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

   if -1.35000000000000003e154 < x < 1.31999999999999998e154

   1. Initial program 69.2%

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

      1. add-exp-log 68.9%

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

      2. pow1/3 68.5%

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

      3. log-pow 68.6%

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

      4. +-commutative 68.6%

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

      5. log1p-udef 68.6%

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

   3. Applied egg-rr 68.6%

      \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]
   4. Step-by-step derivation

      1. *-commutative 68.6%

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

      2. exp-prod 68.5%

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

      3. unpow1/3 68.9%

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

   5. Simplified 68.9%

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

      1. flip3-- 68.9%

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

      2. div-inv 68.9%

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

      3. log1p-udef 68.9%

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

      4. add-exp-log 68.6%

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

      5. rem-cube-cbrt 68.7%

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

      6. rem-cube-cbrt 69.0%

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

   7. Applied egg-rr 69.7%

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

      1. associate-*r/ 69.7%

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

      2. *-rgt-identity 69.7%

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

      3. associate--l+ 99.6%

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

      4. +-inverses 99.6%

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

      5. metadata-eval 99.6%

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

      6. *-commutative 99.6%

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

      7. distribute-lft-in 99.6%

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

      8. *-rgt-identity 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154} \lor \neg \left(x \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x + x \cdot x}\right)}\\ \end{array} \]

Alternative 6: 59.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(1.0 / 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.8%

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

   1. flip3-- 52.8%

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

   2. div-inv 52.8%

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

   3. rem-cube-cbrt 52.7%

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

   4. rem-cube-cbrt 53.2%

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

   5. cbrt-unprod 53.2%

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

   6. pow2 53.2%

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

   7. distribute-rgt-out 53.2%

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

   8. +-commutative 53.2%

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

3. Applied egg-rr 53.2%

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

   1. associate-*r/ 53.2%

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

   2. *-rgt-identity 53.2%

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

   3. +-commutative 53.2%

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

   4. associate--l+ 75.5%

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

   5. +-inverses 75.5%

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

   6. metadata-eval 75.5%

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

   7. +-commutative 75.5%

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

   8. fma-def 75.5%

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

   9. +-commutative 75.5%

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

   10. +-commutative 75.5%

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

5. Simplified 75.5%

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

   1. unpow2 75.5%

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

   2. cbrt-prod 99.2%

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

7. Applied egg-rr 99.2%

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

   1. unpow2 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in x around 0 59.1%

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

    1. fma-udef 59.1%

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

12. Applied egg-rr 59.1%

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

13. Final simplification 59.1%

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

Alternative 7: 53.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. add-exp-log 51.7%

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

   2. pow1/3 51.3%

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

   3. log-pow 51.5%

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

   4. +-commutative 51.5%

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

   5. log1p-udef 51.5%

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

3. Applied egg-rr 51.5%

   \[\leadsto \color{blue}{e^{0.3333333333333333 \cdot \mathsf{log1p}\left(x\right)}} - \sqrt[3]{x} \]
4. Step-by-step derivation

   1. exp-prod 51.5%

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

5. Simplified 51.5%

   \[\leadsto \color{blue}{{\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}} - \sqrt[3]{x} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 51.2%

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

   2. sqrt-unprod 52.1%

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

   3. pow2 52.1%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)} - \sqrt[3]{x}\right)}^{2}}} \]

   4. add-exp-log 52.1%

      \[\leadsto \sqrt{{\left(\color{blue}{e^{\log \left({\left(e^{0.3333333333333333}\right)}^{\left(\mathsf{log1p}\left(x\right)\right)}\right)}} - \sqrt[3]{x}\right)}^{2}} \]

   5. log-pow 52.1%

      \[\leadsto \sqrt{{\left(e^{\color{blue}{\mathsf{log1p}\left(x\right) \cdot \log \left(e^{0.3333333333333333}\right)}} - \sqrt[3]{x}\right)}^{2}} \]

   6. log1p-udef 52.1%

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

   7. add-log-exp 52.1%

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

   8. pow-to-exp 52.1%

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

   9. pow1/3 52.9%

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

7. Applied egg-rr 52.9%

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

   1. unpow2 52.9%

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

   2. rem-sqrt-square 52.9%

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

9. Simplified 52.9%

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

10. Final simplification 52.9%

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

Alternative 8: 53.3% 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 52.8%

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

2. Final simplification 52.8%

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

Alternative 9: 51.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot 0.3333333333333333\right) - \sqrt[3]{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (+ 1.0 (* x 0.3333333333333333)) (cbrt x)))

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in x around 0 50.8%

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

   1. +-commutative 50.8%

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

   2. unpow2 50.8%

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

   3. associate-*r* 50.8%

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

   4. distribute-rgt-out 50.8%

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

   5. *-commutative 50.8%

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

4. Simplified 50.8%

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

5. Taylor expanded in x around 0 51.4%

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

   1. *-commutative 51.4%

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

7. Simplified 51.4%

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

8. Final simplification 51.4%

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

Alternative 10: 3.7% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in x around inf 3.6%

   \[\leadsto \color{blue}{0} \]

3. Final simplification 3.6%

   \[\leadsto 0 \]

Alternative 11: 50.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 52.8%

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

2. Taylor expanded in x around 0 50.0%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 50.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))