2cbrt (problem 3.3.4)

Average Accuracy: 53.8% → 99.2%

Time: 14.5s

Precision: binary64

Cost: 59584

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))) (t_1 (+ 1.0 t_0)))
   (/
    1.0
    (fma
     (cbrt x)
     (+ (cbrt x) (/ (+ -1.0 (pow t_1 3.0)) (+ 1.0 (* t_1 (+ t_0 2.0)))))
     (pow t_0 2.0)))))

C

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

↓

double code(double x) {
	double t_0 = cbrt((1.0 + x));
	double t_1 = 1.0 + t_0;
	return 1.0 / fma(cbrt(x), (cbrt(x) + ((-1.0 + pow(t_1, 3.0)) / (1.0 + (t_1 * (t_0 + 2.0))))), pow(t_0, 2.0));
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	t_1 = Float64(1.0 + t_0)
	return Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + Float64(Float64(-1.0 + (t_1 ^ 3.0)) / Float64(1.0 + Float64(t_1 * Float64(t_0 + 2.0))))), (t_0 ^ 2.0)))
end

Wolfram

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

↓

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + t$95$0), $MachinePrecision]}, N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[(N[(-1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$1 * N[(t$95$0 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 53.8%

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

2. Applied egg-rr 54.9%

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

   [Start] 53.8	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
   flip3-- [=>] 53.9	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   div-inv [=>] 53.9	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   rem-cube-cbrt [=>] 53.8	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   rem-cube-cbrt [=>] 54.9	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   pow2 [=>] 54.9	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{2}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   distribute-rgt-out [=>] 54.9	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
   +-commutative [<=] 54.9	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

3. Simplified 99.2%

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

   [Start] 54.9	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 54.9	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 54.9	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 54.9	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 99.2	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 99.2	\[ \frac{1 + \color{blue}{0}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{\color{blue}{1}}{{\left(\sqrt[3]{x + 1}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {\left(\sqrt[3]{x + 1}\right)}^{2}}} \]
   fma-def [=>] 99.2	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)}} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, {\left(\sqrt[3]{x + 1}\right)}^{2}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2}\right)} \]

4. Applied egg-rr 74.2%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   Proof

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   expm1-log1p-u [=>] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

5. Applied egg-rr 99.2%

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

   [Start] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)\right) + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   expm1-udef [=>] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} - 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   flip3-- [=>] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\frac{{\left(e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   log1p-udef [=>] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(e^{\color{blue}{\log \left(1 + \sqrt[3]{1 + x}\right)}}\right)}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   add-exp-log [<=] 73.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\color{blue}{\left(1 + \sqrt[3]{1 + x}\right)}}^{3} - {1}^{3}}{e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   metadata-eval [=>] 73.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - \color{blue}{1}}{e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   log1p-udef [=>] 73.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{e^{\color{blue}{\log \left(1 + \sqrt[3]{1 + x}\right)}} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   add-exp-log [<=] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{\color{blue}{\left(1 + \sqrt[3]{1 + x}\right)} \cdot e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   log1p-udef [=>] 74.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{\left(1 + \sqrt[3]{1 + x}\right) \cdot e^{\color{blue}{\log \left(1 + \sqrt[3]{1 + x}\right)}} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   add-exp-log [<=] 74.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \color{blue}{\left(1 + \sqrt[3]{1 + x}\right)} + \left(1 \cdot 1 + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   metadata-eval [=>] 74.9	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right) + \left(\color{blue}{1} + e^{\mathsf{log1p}\left(\sqrt[3]{1 + x}\right)} \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

6. Simplified 99.2%

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

   [Start] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} - 1}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right) + \left(1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   sub-neg [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} + \left(-1\right)}}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right) + \left(1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{{\left(1 + \sqrt[3]{1 + x}\right)}^{3} + \color{blue}{-1}}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right) + \left(1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{\color{blue}{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right) + \left(1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot 1\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   +-commutative [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{\color{blue}{\left(1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot 1\right) + \left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right)}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   *-rgt-identity [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{\left(1 + \color{blue}{\left(1 + \sqrt[3]{1 + x}\right)}\right) + \left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   associate-+l+ [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{\color{blue}{1 + \left(\left(1 + \sqrt[3]{1 + x}\right) + \left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right)\right)}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   *-rgt-identity [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{1 + \left(\color{blue}{\left(1 + \sqrt[3]{1 + x}\right) \cdot 1} + \left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \sqrt[3]{1 + x}\right)\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   distribute-lft-in [<=] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{1 + \color{blue}{\left(1 + \sqrt[3]{1 + x}\right) \cdot \left(1 + \left(1 + \sqrt[3]{1 + x}\right)\right)}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   associate-+r+ [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot \color{blue}{\left(\left(1 + 1\right) + \sqrt[3]{1 + x}\right)}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
   metadata-eval [=>] 99.2	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \frac{-1 + {\left(1 + \sqrt[3]{1 + x}\right)}^{3}}{1 + \left(1 + \sqrt[3]{1 + x}\right) \cdot \left(\color{blue}{2} + \sqrt[3]{1 + x}\right)} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

7. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	39168

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

Alternative 2
Accuracy	88.7%
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ t_2 := \sqrt[3]{x} \cdot t_1\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + e^{\mathsf{log1p}\left(x\right) \cdot 0.6666666666666666}}\\ \end{array} \]

Alternative 3
Accuracy	61.2%
Cost	33092

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	33024

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

Alternative 5
Accuracy	99.2%
Cost	32896

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

Alternative 6
Accuracy	88.6%
Cost	26888

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_1}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{x}}{\frac{1}{t_1}} + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 7
Accuracy	88.6%
Cost	26824

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} + t_0\\ t_2 := \sqrt[3]{x} \cdot t_1\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2 + {\left(1 + x\right)}^{0.6666666666666666}}\\ \end{array} \]

Alternative 8
Accuracy	87.6%
Cost	26760

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

Alternative 9
Accuracy	78.6%
Cost	26696

\[\begin{array}{l} t_0 := \sqrt[3]{x \cdot x}\\ t_1 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\\ t_2 := \frac{1}{1 + t_1}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{1}{t_1 + t_0}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \left(t_0 + \sqrt[3]{x + x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	56.9%
Cost	26308

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	69.5%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.4 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x + x \cdot x}\right)}\\ \end{array} \]

Alternative 12
Accuracy	61.3%
Cost	20232

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+15}:\\ \;\;\;\;t_0 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(1 + x\right)}^{0.6666666666666666} + \sqrt[3]{x} \cdot \left(1 + \sqrt[3]{x}\right)}\\ \end{array} \]

Alternative 13
Accuracy	60.2%
Cost	19656

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+15}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \sqrt[3]{x}\right)\right)}^{-2}\\ \end{array} \]

Alternative 14
Accuracy	60.2%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+15} \lor \neg \left(x \leq 2.9 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{1}{1 + {\left(\sqrt[3]{x}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array} \]

Alternative 15
Accuracy	52.7%
Cost	6848

\[\frac{1}{1 + \sqrt[3]{x \cdot x}} \]

Alternative 16
Accuracy	3.6%
Cost	64

\[0 \]

Alternative 17
Accuracy	50.3%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))