2cbrt (problem 3.3.4)

?

Percentage Accurate: 54.1% → 99.2%
Time: 9.7s
Precision: binary64
Cost: 52228

?

\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (if (<= (- t_0 (cbrt x)) 0.0)
     (/ 1.0 (fma (cbrt x) (+ (cbrt x) (cbrt x)) (pow t_0 2.0)))
     (/
      (- (+ 1.0 x) x)
      (+ (cbrt (pow (+ 1.0 x) 2.0)) (* (cbrt x) (+ (cbrt x) t_0)))))))
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 ((t_0 - cbrt(x)) <= 0.0) {
		tmp = 1.0 / fma(cbrt(x), (cbrt(x) + cbrt(x)), pow(t_0, 2.0));
	} else {
		tmp = ((1.0 + x) - x) / (cbrt(pow((1.0 + x), 2.0)) + (cbrt(x) * (cbrt(x) + t_0)));
	}
	return tmp;
}

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 (Float64(t_0 - cbrt(x)) <= 0.0)
		tmp = Float64(1.0 / fma(cbrt(x), Float64(cbrt(x) + cbrt(x)), (t_0 ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 + x) - x) / Float64(cbrt((Float64(1.0 + x) ^ 2.0)) + Float64(cbrt(x) * Float64(cbrt(x) + t_0))));
	end
	return tmp
end

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[LessEqual[N[(t$95$0 - N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision], 0.0], 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[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * N[(N[Power[x, 1/3], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\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}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 9 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Derivation?

  1. Split input into 2 regimes

  2. if (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x)) < 0.0

    1. Initial program 4.1%

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

    2. Applied egg-rr4.1%

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

      [Start]4.1%

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

      flip3-- [=>]4.1%

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

      div-inv [=>]4.1%

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

      rem-cube-cbrt [=>]3.9%

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

      rem-cube-cbrt [=>]4.1%

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

      cbrt-unprod [=>]4.1%

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

      pow2 [=>]4.1%

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

      distribute-rgt-out [=>]4.1%

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

      +-commutative [<=]4.1%

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

    3. Simplified49.5%

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

      [Start]4.1%

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

      associate-*r/ [=>]4.1%

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

      *-rgt-identity [=>]4.1%

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

      +-commutative [=>]4.1%

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

      associate--l+ [=>]49.5%

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

      +-inverses [=>]49.5%

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

      metadata-eval [=>]49.5%

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

      +-commutative [=>]49.5%

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

      fma-def [=>]49.5%

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

      +-commutative [=>]49.5%

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

      +-commutative [=>]49.5%

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

    4. Applied egg-rr94.1%

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

      [Start]49.5%

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

      +-commutative [=>]49.5%

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

      expm1-log1p-u [=>]47.7%

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

      expm1-udef [=>]47.7%

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

      pow1/3 [=>]47.3%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(x + 1\right)}^{2}\right)}^{0.3333333333333333}}\right)} - 1\right)} \]

      +-commutative [<=]47.3%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left({\left({\color{blue}{\left(1 + x\right)}}^{2}\right)}^{0.3333333333333333}\right)} - 1\right)} \]

      unpow2 [=>]47.3%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left({\color{blue}{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right)}}^{0.3333333333333333}\right)} - 1\right)} \]

      pow-prod-down [<=]42.2%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(\color{blue}{{\left(1 + x\right)}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}}\right)} - 1\right)} \]

      +-commutative [=>]42.2%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left({\color{blue}{\left(x + 1\right)}}^{0.3333333333333333} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} - 1\right)} \]

      pow1/3 [<=]42.4%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(\color{blue}{\sqrt[3]{x + 1}} \cdot {\left(1 + x\right)}^{0.3333333333333333}\right)} - 1\right)} \]

      +-commutative [=>]42.4%

      \[ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{1 + x} + \sqrt[3]{x}, e^{\mathsf{log1p}\left(\sqrt[3]{x + 1} \cdot {\color{blue}{\left(x + 1\right)}}^{0.3333333333333333}\right)} - 1\right)} \]

      pow1/3 [<=]94.1%

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

      pow2 [=>]94.1%

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

      +-commutative [<=]94.1%

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

    5. Simplified98.6%

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

      [Start]94.1%

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

      expm1-def [=>]94.1%

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

      expm1-log1p [=>]98.6%

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

    6. Applied egg-rr98.5%

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

      [Start]98.6%

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

      +-commutative [=>]98.6%

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

      add-cube-cbrt [=>]98.4%

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

      pow3 [=>]98.5%

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

      +-commutative [<=]98.5%

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

    7. Taylor expanded in x around inf 42.6%

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

    8. Simplified98.6%

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

      [Start]42.6%

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

      unpow1/3 [=>]98.6%

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

    if 0.0 < (-.f64 (cbrt.f64 (+.f64 x 1)) (cbrt.f64 x))

    1. Initial program 98.3%

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

    2. Applied egg-rr99.9%

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

      [Start]98.3%

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

      flip3-- [=>]98.2%

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

      rem-cube-cbrt [=>]98.1%

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

      rem-cube-cbrt [=>]99.8%

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

      cbrt-unprod [=>]99.9%

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

      pow2 [=>]99.9%

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

      distribute-rgt-out [=>]99.9%

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

      +-commutative [<=]99.9%

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

  4. Final simplification99.2%

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

Alternatives

Alternative 1
Accuracy99.2%
Cost52228
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \mathbf{if}\;t_0 - \sqrt[3]{x} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{x}, {t_0}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\sqrt[3]{{\left(1 + x\right)}^{2}} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x} + t_0\right)}\\ \end{array} \]
Alternative 2
Accuracy75.7%
Cost45892
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ t_1 := t_0 - \sqrt[3]{x}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Accuracy99.2%
Cost39168
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + t_0, {t_0}^{2}\right)} \end{array} \]
Alternative 4
Accuracy87.6%
Cost33032
\[\begin{array}{l} t_0 := \sqrt[3]{x} + \sqrt[3]{1 + x}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, 1\right)}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, \sqrt[3]{x \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(1 + x\right)}^{0.6666666666666666}\right)}\\ \end{array} \]
Alternative 5
Accuracy59.7%
Cost32640
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, 1 + \sqrt[3]{x}, {\left(\sqrt[3]{1 + x}\right)}^{2}\right)} \]
Alternative 6
Accuracy59.4%
Cost26176
\[\frac{1}{\mathsf{fma}\left(\sqrt[3]{x}, \sqrt[3]{x} + \sqrt[3]{1 + x}, 1\right)} \]
Alternative 7
Accuracy54.1%
Cost13120
\[\sqrt[3]{1 + x} - \sqrt[3]{x} \]
Alternative 8
Accuracy3.6%
Cost64
\[0 \]
Alternative 9
Accuracy50.4%
Cost64
\[1 \]

Reproduce?

herbie shell --seed 2023178 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))