2cbrt (problem 3.3.4)

Percentage Accurate: 6.9% → 99.5%

Time: 10.6s

Alternatives: 7

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Initial Program: 6.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. flip3-- 7.0%

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

   2. div-inv 7.0%

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

   3. rem-cube-cbrt 5.8%

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

   4. rem-cube-cbrt 8.6%

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

   5. +-commutative 8.6%

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

   6. distribute-rgt-out 8.6%

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

   7. +-commutative 8.6%

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

   8. fma-define 8.6%

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

   9. add-exp-log 8.6%

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

4. Applied egg-rr 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-rgt-identity 8.6%

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

   3. +-commutative 8.6%

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

   4. associate--l+ 93.2%

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

   5. +-inverses 93.2%

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

   6. metadata-eval 93.2%

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

   7. +-commutative 93.2%

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

   8. exp-prod 92.2%

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

6. Simplified 92.2%

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

   1. add-sqr-sqrt 92.2%

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

   2. unpow-prod-down 93.9%

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

8. Applied egg-rr 93.9%

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

   1. pow2 93.9%

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

   2. add-exp-log 93.4%

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

   3. log-pow 93.2%

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

   4. log1p-undefine 93.2%

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

   5. pow1/2 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}\right)}^{2}\right)} \]

   6. log-pow 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)}}\right)}^{2}\right)} \]

   7. rem-log-exp 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \left(0.5 \cdot \color{blue}{0.6666666666666666}\right)}\right)}^{2}\right)} \]

   8. metadata-eval 93.2%

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

   9. pow-to-exp 92.9%

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

   10. pow1/3 98.5%

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

   11. +-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

    1. flip3-+ 98.4%

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

    2. +-commutative 98.4%

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

    3. rem-cube-cbrt 98.9%

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

    4. rem-cube-cbrt 99.5%

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

    5. +-commutative 99.5%

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

    6. +-commutative 99.5%

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

    7. +-commutative 99.5%

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

    8. pow2 99.5%

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

    9. +-commutative 99.5%

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

    10. distribute-rgt-out-- 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. flip3-- 7.0%

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

   2. div-inv 7.0%

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

   3. rem-cube-cbrt 5.8%

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

   4. rem-cube-cbrt 8.6%

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

   5. +-commutative 8.6%

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

   6. distribute-rgt-out 8.6%

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

   7. +-commutative 8.6%

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

   8. fma-define 8.6%

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

   9. add-exp-log 8.6%

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

4. Applied egg-rr 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-rgt-identity 8.6%

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

   3. +-commutative 8.6%

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

   4. associate--l+ 93.2%

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

   5. +-inverses 93.2%

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

   6. metadata-eval 93.2%

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

   7. +-commutative 93.2%

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

   8. exp-prod 92.2%

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

6. Simplified 92.2%

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

   1. add-sqr-sqrt 92.2%

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

   2. unpow-prod-down 93.9%

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

8. Applied egg-rr 93.9%

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

   1. pow2 93.9%

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

   2. add-exp-log 93.4%

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

   3. log-pow 93.2%

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

   4. log1p-undefine 93.2%

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

   5. pow1/2 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}\right)}^{2}\right)} \]

   6. log-pow 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)}}\right)}^{2}\right)} \]

   7. rem-log-exp 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \left(0.5 \cdot \color{blue}{0.6666666666666666}\right)}\right)}^{2}\right)} \]

   8. metadata-eval 93.2%

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

   9. pow-to-exp 92.9%

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

   10. pow1/3 98.5%

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

   11. +-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 98.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

   1. flip3-- 7.0%

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

   2. div-inv 7.0%

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

   3. rem-cube-cbrt 5.8%

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

   4. rem-cube-cbrt 8.6%

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

   5. +-commutative 8.6%

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

   6. distribute-rgt-out 8.6%

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

   7. +-commutative 8.6%

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

   8. fma-define 8.6%

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

   9. add-exp-log 8.6%

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

4. Applied egg-rr 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-rgt-identity 8.6%

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

   3. +-commutative 8.6%

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

   4. associate--l+ 93.2%

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

   5. +-inverses 93.2%

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

   6. metadata-eval 93.2%

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

   7. +-commutative 93.2%

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

   8. exp-prod 92.2%

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

6. Simplified 92.2%

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

   1. add-sqr-sqrt 92.2%

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

   2. unpow-prod-down 93.9%

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

8. Applied egg-rr 93.9%

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

   1. pow2 93.9%

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

   2. add-exp-log 93.4%

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

   3. log-pow 93.2%

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

   4. log1p-undefine 93.2%

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

   5. pow1/2 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}\right)}^{2}\right)} \]

   6. log-pow 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)}}\right)}^{2}\right)} \]

   7. rem-log-exp 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \left(0.5 \cdot \color{blue}{0.6666666666666666}\right)}\right)}^{2}\right)} \]

   8. metadata-eval 93.2%

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

   9. pow-to-exp 92.9%

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

   10. pow1/3 98.5%

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

   11. +-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

    1. fma-undefine 98.4%

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

    2. +-commutative 98.4%

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

12. Applied egg-rr 98.4%

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

13. Final simplification 98.4%

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

Alternative 4: 96.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) * 2.0), (cbrt(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] * 2.0), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

   1. flip3-- 7.0%

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

   2. div-inv 7.0%

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

   3. rem-cube-cbrt 5.8%

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

   4. rem-cube-cbrt 8.6%

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

   5. +-commutative 8.6%

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

   6. distribute-rgt-out 8.6%

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

   7. +-commutative 8.6%

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

   8. fma-define 8.6%

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

   9. add-exp-log 8.6%

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

4. Applied egg-rr 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-rgt-identity 8.6%

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

   3. +-commutative 8.6%

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

   4. associate--l+ 93.2%

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

   5. +-inverses 93.2%

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

   6. metadata-eval 93.2%

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

   7. +-commutative 93.2%

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

   8. exp-prod 92.2%

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

6. Simplified 92.2%

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

   1. add-sqr-sqrt 92.2%

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

   2. unpow-prod-down 93.9%

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

8. Applied egg-rr 93.9%

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

   1. pow2 93.9%

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

   2. add-exp-log 93.4%

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

   3. log-pow 93.2%

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

   4. log1p-undefine 93.2%

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

   5. pow1/2 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \log \color{blue}{\left({\left(e^{0.6666666666666666}\right)}^{0.5}\right)}}\right)}^{2}\right)} \]

   6. log-pow 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \color{blue}{\left(0.5 \cdot \log \left(e^{0.6666666666666666}\right)\right)}}\right)}^{2}\right)} \]

   7. rem-log-exp 93.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(e^{\log \left(1 + x\right) \cdot \left(0.5 \cdot \color{blue}{0.6666666666666666}\right)}\right)}^{2}\right)} \]

   8. metadata-eval 93.2%

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

   9. pow-to-exp 92.9%

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

   10. pow1/3 98.5%

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

   11. +-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

11. Taylor expanded in x around inf 97.1%

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

12. Final simplification 97.1%

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

Alternative 5: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt[3]{x}}\\ t\_0 \cdot \left(t\_0 \cdot 0.3333333333333333\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (cbrt x)))) (* t_0 (* t_0 0.3333333333333333))))

C

double code(double x) {
	double t_0 = 1.0 / cbrt(x);
	return t_0 * (t_0 * 0.3333333333333333);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.cbrt(x);
	return t_0 * (t_0 * 0.3333333333333333);
}

Julia

function code(x)
	t_0 = Float64(1.0 / cbrt(x))
	return Float64(t_0 * Float64(t_0 * 0.3333333333333333))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.4%

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

   1. add-cube-cbrt 6.4%

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

   2. pow3 6.4%

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

   3. pow-to-exp 6.4%

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

4. Applied egg-rr 6.4%

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

5. Taylor expanded in x around inf 46.0%

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

   1. exp-to-pow 48.5%

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

   2. pow3 48.5%

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

   3. add-cube-cbrt 49.1%

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

   4. *-commutative 49.1%

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

   5. add-sqr-sqrt 49.0%

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

   6. associate-*l* 49.0%

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

   7. cbrt-div 49.2%

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

   8. metadata-eval 49.2%

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

   9. unpow2 49.2%

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

   10. cbrt-prod 49.0%

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

   11. sqrt-div 49.0%

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

   12. metadata-eval 49.0%

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

   13. sqrt-unprod 49.0%

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

   14. add-sqr-sqrt 49.0%

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

7. Applied egg-rr 96.9%

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

Alternative 6: 50.6% 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 6.4%

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

   1. add-cube-cbrt 6.4%

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

   2. pow3 6.4%

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

   3. pow-to-exp 6.4%

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

4. Applied egg-rr 6.4%

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

5. Taylor expanded in x around inf 46.0%

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

   1. exp-to-pow 48.5%

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

   2. pow3 48.5%

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

   3. add-cube-cbrt 49.1%

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

   4. *-commutative 49.1%

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

   5. pow-flip 52.2%

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

   6. metadata-eval 52.2%

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

7. Applied egg-rr 52.2%

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

8. Final simplification 52.2%

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

Alternative 7: 5.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

3. Taylor expanded in x around 0 1.8%

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

   1. sub-neg 1.8%

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

   2. rem-square-sqrt 0.0%

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

   3. fabs-sqr 0.0%

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

   4. rem-square-sqrt 5.2%

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

   5. fabs-neg 5.2%

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

   6. unpow1/3 5.2%

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

   7. metadata-eval 5.2%

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

   8. pow-sqr 5.2%

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

   9. fabs-sqr 5.2%

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

   10. pow-sqr 5.2%

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

   11. metadata-eval 5.2%

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

   12. unpow1/3 5.2%

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

5. Simplified 5.2%

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

Developer Target 1: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024116 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (cbrt (+ x 1)) (cbrt (+ x 1))) (* (cbrt x) (cbrt (+ x 1))) (* (cbrt x) (cbrt x)))))

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