2cbrt (problem 3.3.4)

Percentage Accurate: 53.9% → 99.6%

Time: 12.5s

Precision: binary64

Cost: 59200

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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[(x + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 55.8%

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

2. Applied egg-rr 57.3%

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

   [Start] 55.8%	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
   flip3-- [=>] 56.2%	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   div-inv [=>] 56.2%	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
   rem-cube-cbrt [=>] 55.8%	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   rem-cube-cbrt [=>] 57.2%	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   cbrt-unprod [=>] 57.3%	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(x + 1\right) \cdot \left(x + 1\right)}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   pow2 [=>] 57.3%	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{{\left(x + 1\right)}^{2}}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
   distribute-rgt-out [=>] 57.3%	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
   +-commutative [<=] 57.3%	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

3. Simplified 75.1%

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

   [Start] 57.3%	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate-*r/ [=>] 57.3%	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
   *-rgt-identity [=>] 57.3%	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 57.3%	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   associate--l+ [=>] 75.1%	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-inverses [=>] 75.1%	\[ \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   metadata-eval [=>] 75.1%	\[ \frac{\color{blue}{1}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
   +-commutative [=>] 75.1%	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(x + 1\right)}^{2}}}} \]
   fma-def [=>] 75.1%	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)}} \]
   +-commutative [=>] 75.1%	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)} \]
   +-commutative [=>] 75.1%	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\color{blue}{\left(1 + x\right)}}^{2}}\right)} \]

4. Applied egg-rr 99.1%

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

   [Start] 75.1%	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]
   fma-udef [=>] 75.1%	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}}} \]
   unpow2 [=>] 75.1%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{\left(1 + x\right) \cdot \left(1 + x\right)}}} \]
   cbrt-prod [=>] 99.1%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \color{blue}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}} \]
   add-sqr-sqrt [=>] 74.5%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}} \cdot \sqrt[3]{1 + x}} \]
   cbrt-unprod [<=] 74.6%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \color{blue}{\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)} \cdot \sqrt[3]{1 + x}} \]
   add-sqr-sqrt [=>] 74.6%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \sqrt[3]{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}} \]
   cbrt-unprod [<=] 74.6%	\[ \frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)}} \]
   +-commutative [=>] 74.6%	\[ \frac{1}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right) + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   pow2 [=>] 74.6%	\[ \frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   cbrt-unprod [=>] 74.5%	\[ \frac{1}{{\color{blue}{\left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{1 + x}}\right)}}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]
   add-sqr-sqrt [<=] 99.1%	\[ \frac{1}{{\left(\sqrt[3]{\color{blue}{1 + x}}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)} \]

5. Applied egg-rr 98.9%

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

   [Start] 99.1%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)} \]
   add-cube-cbrt [=>] 98.9%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}}} \]
   pow3 [=>] 98.9%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{{\left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\right)}^{3}}} \]
   +-commutative [=>] 98.9%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + {\left(\sqrt[3]{\sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}}\right)}^{3}} \]

6. Applied egg-rr 89.3%

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

   [Start] 98.9%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + {\left(\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}\right)}^{3}} \]
   rem-cube-cbrt [=>] 99.1%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)}} \]
   +-commutative [<=] 99.1%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}} \]
   flip3-+ [=>] 99.1%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \sqrt[3]{x} \cdot \color{blue}{\frac{{\left(\sqrt[3]{x}\right)}^{3} + {\left(\sqrt[3]{1 + x}\right)}^{3}}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}}} \]
   associate-*r/ [=>] 88.9%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\frac{\sqrt[3]{x} \cdot \left({\left(\sqrt[3]{x}\right)}^{3} + {\left(\sqrt[3]{1 + x}\right)}^{3}\right)}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}}} \]
   rem-cube-cbrt [=>] 89.1%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(\color{blue}{x} + {\left(\sqrt[3]{1 + x}\right)}^{3}\right)}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}} \]
   rem-cube-cbrt [=>] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \color{blue}{\left(1 + x\right)}\right)}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}} \]
   +-commutative [=>] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \color{blue}{\left(x + 1\right)}\right)}{\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}} \]
   unpow2 [<=] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{2}} + \left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x} - \sqrt[3]{x} \cdot \sqrt[3]{1 + x}\right)}} \]
   distribute-rgt-out-- [=>] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(\sqrt[3]{x}\right)}^{2} + \color{blue}{\sqrt[3]{1 + x} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}}} \]
   +-commutative [=>] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{\color{blue}{x + 1}} \cdot \left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}} \]

7. Simplified 99.6%

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

   [Start] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\sqrt[3]{x} \cdot \left(x + \left(x + 1\right)\right)}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}} \]
   *-commutative [=>] 89.3%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{\color{blue}{\left(x + \left(x + 1\right)\right) \cdot \sqrt[3]{x}}}{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}} \]
   associate-/l* [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \color{blue}{\frac{x + \left(x + 1\right)}{\frac{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}{\sqrt[3]{x}}}}} \]
   +-commutative [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{x + \color{blue}{\left(1 + x\right)}}{\frac{{\left(\sqrt[3]{x}\right)}^{2} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right)}{\sqrt[3]{x}}}} \]
   +-commutative [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{x + \left(1 + x\right)}{\frac{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} - \sqrt[3]{x}\right) + {\left(\sqrt[3]{x}\right)}^{2}}}{\sqrt[3]{x}}}} \]
   fma-def [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{x + \left(1 + x\right)}{\frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x + 1}, \sqrt[3]{x + 1} - \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}}{\sqrt[3]{x}}}} \]
   +-commutative [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{x + \left(1 + x\right)}{\frac{\mathsf{fma}\left(\sqrt[3]{\color{blue}{1 + x}}, \sqrt[3]{x + 1} - \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\sqrt[3]{x}}}} \]
   +-commutative [=>] 99.6%	\[ \frac{1}{{\left(\sqrt[3]{1 + x}\right)}^{2} + \frac{x + \left(1 + x\right)}{\frac{\mathsf{fma}\left(\sqrt[3]{1 + x}, \sqrt[3]{\color{blue}{1 + x}} - \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}{\sqrt[3]{x}}}} \]

8. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	59200

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

Alternative 2
Accuracy	99.6%
Cost	52928

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

Alternative 3
Accuracy	61.3%
Cost	39108

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

Alternative 4
Accuracy	88.7%
Cost	33160

\[\begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_0}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t_0 + \sqrt[3]{{\left(1 + x\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 5
Accuracy	88.8%
Cost	33096

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + t_1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 6
Accuracy	61.3%
Cost	33092

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

Alternative 7
Accuracy	88.7%
Cost	33032

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

Alternative 8
Accuracy	60.2%
Cost	32900

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	99.2%
Cost	32896

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

Alternative 10
Accuracy	79.6%
Cost	26888

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 + \sqrt[3]{x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x} \cdot t_1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{{t_0}^{2} + \left(\sqrt[3]{x \cdot x} + \sqrt[3]{x \cdot \left(1 + x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + t_1 \cdot e^{\log x \cdot 0.3333333333333333}}\\ \end{array} \]

Alternative 11
Accuracy	53.9%
Cost	13120

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

Alternative 12
Accuracy	51.3%
Cost	6848

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

Alternative 13
Accuracy	52.6%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 0.62:\\ \;\;\;\;1 - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x \cdot 0.6666666666666666}\\ \end{array} \]

Alternative 14
Accuracy	3.6%
Cost	64

\[0 \]

Alternative 15
Accuracy	50.2%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023229 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))