2cbrt (problem 3.3.4)

Average Accuracy: 53.4% → 99.3%

Time: 10.7s

Precision: binary64

Cost: 39433

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+154} \lor \neg \left(x \leq 10^{+151}\right):\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (or (<= x -2e+154) (not (<= x 1e+151)))
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt x)) (pow t_0 2.0)))
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) t_0) (cbrt (pow (+ 1.0 x) 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 tmp;
	if ((x <= -2e+154) || !(x <= 1e+151)) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt(x)), pow(t_0, 2.0));
	} else {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + t_0), cbrt(pow((1.0 + x), 2.0)));
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = cbrt(Float64(1.0 + x))
	tmp = 0.0
	if ((x <= -2e+154) || !(x <= 1e+151))
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(x)), (t_0 ^ 2.0)));
	else
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + t_0), cbrt((Float64(1.0 + x) ^ 2.0))));
	end
	return tmp
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]}, If[Or[LessEqual[x, -2e+154], N[Not[LessEqual[x, 1e+151]], $MachinePrecision]], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] + N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000007e154 or 1.00000000000000002e151 < x

   1. Initial program 4.7%

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

   2. Applied egg-rr 4.7%

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

      [Start] 4.7	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
      flip3-- [=>] 4.7	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
      div-inv [=>] 4.7	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
      rem-cube-cbrt [=>] 3.3	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      rem-cube-cbrt [=>] 4.7	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      cbrt-unprod [=>] 4.7	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(x + 1\right) \cdot \left(x + 1\right)}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      pow2 [=>] 4.7	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{{\left(x + 1\right)}^{2}}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      distribute-rgt-out [=>] 4.7	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
      +-commutative [<=] 4.7	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   3. Simplified 5.8%

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

      [Start] 4.7	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate-*r/ [=>] 4.7	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
      *-rgt-identity [=>] 4.7	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 4.7	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate--l+ [=>] 5.8	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-inverses [=>] 5.8	\[ \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      metadata-eval [=>] 5.8	\[ \frac{\color{blue}{1}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 5.8	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(x + 1\right)}^{2}}}} \]
      fma-def [=>] 5.8	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)}} \]
      +-commutative [=>] 5.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)} \]
      +-commutative [=>] 5.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\color{blue}{\left(1 + x\right)}}^{2}}\right)} \]

   4. Applied egg-rr 98.4%

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

      [Start] 5.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)} \]
      unpow2 [=>] 5.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{\color{blue}{\left(1 + x\right) \cdot \left(1 + x\right)}}\right)} \]
      cbrt-prod [=>] 98.4	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right)} \]

   5. Simplified 98.4%

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

      [Start] 98.4	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right)} \]
      unpow2 [<=] 98.4	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \color{blue}{{\left(\sqrt[3]{1 + x}\right)}^{2}}\right)} \]

   6. Applied egg-rr 98.3%

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

      [Start] 98.4	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
      add-cube-cbrt [=>] 98.3	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\left(\sqrt[3]{\sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + x}}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
      pow3 [=>] 98.3	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{{\left(\sqrt[3]{\sqrt[3]{1 + x}}\right)}^{3}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

   7. Taylor expanded in x around inf 46.8%

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

   8. Simplified 98.4%

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

      [Start] 46.8	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, {x}^{0.3333333333333333} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
      unpow1/3 [=>] 98.4	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \color{blue}{\sqrt[3]{x}} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]

   if -2.00000000000000007e154 < x < 1.00000000000000002e151

   1. Initial program 69.8%

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

   2. Applied egg-rr 71.3%

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

      [Start] 69.8	\[ \sqrt[3]{x + 1} - \sqrt[3]{x} \]
      flip3-- [=>] 70.0	\[ \color{blue}{\frac{{\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
      div-inv [=>] 70.0	\[ \color{blue}{\left({\left(\sqrt[3]{x + 1}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}} \]
      rem-cube-cbrt [=>] 70.3	\[ \left(\color{blue}{\left(x + 1\right)} - {\left(\sqrt[3]{x}\right)}^{3}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      rem-cube-cbrt [=>] 71.3	\[ \left(\left(x + 1\right) - \color{blue}{x}\right) \cdot \frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      cbrt-unprod [=>] 71.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\color{blue}{\sqrt[3]{\left(x + 1\right) \cdot \left(x + 1\right)}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      pow2 [=>] 71.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{\color{blue}{{\left(x + 1\right)}^{2}}} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)} \]
      distribute-rgt-out [=>] 71.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{x + 1}\right)}} \]
      +-commutative [<=] 71.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]

   3. Simplified 99.5%

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

      [Start] 71.3	\[ \left(\left(x + 1\right) - x\right) \cdot \frac{1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate-*r/ [=>] 71.3	\[ \color{blue}{\frac{\left(\left(x + 1\right) - x\right) \cdot 1}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}} \]
      *-rgt-identity [=>] 71.3	\[ \frac{\color{blue}{\left(x + 1\right) - x}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 71.3	\[ \frac{\color{blue}{\left(1 + x\right)} - x}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      associate--l+ [=>] 99.5	\[ \frac{\color{blue}{1 + \left(x - x\right)}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-inverses [=>] 99.5	\[ \frac{1 + \color{blue}{0}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      metadata-eval [=>] 99.5	\[ \frac{\color{blue}{1}}{\sqrt[3]{{\left(x + 1\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{1}{\color{blue}{\sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(x + 1\right)}^{2}}}} \]
      fma-def [=>] 99.5	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x + 1} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)}} \]
      +-commutative [=>] 99.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{\color{blue}{1 + x}} + \sqrt[3]{x}, \sqrt[3]{{\left(x + 1\right)}^{2}}\right)} \]
      +-commutative [=>] 99.5	\[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{{\color{blue}{\left(1 + x\right)}}^{2}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+154} \lor \neg \left(x \leq 10^{+151}\right):\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, \sqrt[3]{{\left(1 + x\right)}^{2}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.2%
Cost	52228

\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \end{array} \]

Alternative 2
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 3
Accuracy	77.9%
Cost	32964

\[\begin{array}{l} t_0 := \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)\\ \mathbf{if}\;x \leq 10^{-41}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 + e^{0.6666666666666666 \cdot \mathsf{log1p}\left(x\right)}}\\ \end{array} \]

Alternative 4
Accuracy	77.8%
Cost	32900

\[\begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq 10^{-41}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(\sqrt[3]{x} \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]

Alternative 5
Accuracy	59.1%
Cost	32640

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

Alternative 6
Accuracy	58.9%
Cost	32512

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

Alternative 7
Accuracy	58.9%
Cost	19904

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

Alternative 8
Accuracy	59.8%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+15}:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+15}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 9
Accuracy	57.2%
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq -0.24:\\ \;\;\;\;{\left(\sqrt[3]{x}\right)}^{-2}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 10
Accuracy	55.6%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -0.24:\\ \;\;\;\;{\left(x \cdot x\right)}^{-0.3333333333333333}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{1}{1 + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 11
Accuracy	52.8%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666}}\\ \end{array} \]

Alternative 12
Accuracy	3.6%
Cost	64

\[0 \]

Alternative 13
Accuracy	50.0%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))