Average Error: 29.9 → 0.6
Time: 5.4s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1 + \left(x - x\right)}{\sqrt[3]{{t_0}^{2}} \cdot \left(t_0 \cdot \sqrt[3]{t_0}\right) + \mathsf{fma}\left(\sqrt[3]{x}, t_0, {\left(\sqrt[3]{x}\right)}^{2}\right)} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/
    (+ 1.0 (- x x))
    (+
     (* (cbrt (pow t_0 2.0)) (* t_0 (cbrt t_0)))
     (fma (cbrt x) t_0 (pow (cbrt x) 2.0))))))
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 + (x - x)) / ((cbrt(pow(t_0, 2.0)) * (t_0 * cbrt(t_0))) + fma(cbrt(x), t_0, pow(cbrt(x), 2.0)));
}

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(Float64(1.0 + Float64(x - x)) / Float64(Float64(cbrt((t_0 ^ 2.0)) * Float64(t_0 * cbrt(t_0))) + fma(cbrt(x), t_0, (cbrt(x) ^ 2.0))))
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]}, N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$0 * N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 1/3], $MachinePrecision] * t$95$0 + N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

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

Error

Bits error versus x

Derivation

  1. Initial program 29.9

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

  2. Applied egg-rr29.3

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

  3. Applied egg-rr15.5

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

  4. Applied egg-rr0.6

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

  5. Applied egg-rr0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))