Average Error: 29.5 → 0.5
Time: 6.4s
Precision: binary64
Cost: 45632
\[\sqrt[3]{x + 1} - \sqrt[3]{x} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \mathsf{fma}\left(t_0, \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)} \end{array} \]
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (+ 1.0 x))))
   (/ 1.0 (+ (pow t_0 2.0) (fma t_0 (cbrt x) (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 / (pow(t_0, 2.0) + fma(t_0, cbrt(x), 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(1.0 / Float64((t_0 ^ 2.0) + fma(t_0, cbrt(x), (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[(1.0 / N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$0 * N[Power[x, 1/3], $MachinePrecision] + 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}{{t_0}^{2} + \mathsf{fma}\left(t_0, \sqrt[3]{x}, {\left(\sqrt[3]{x}\right)}^{2}\right)}
\end{array}

Error

Derivation

  1. Initial program 29.5

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

  2. Applied egg-rr28.8

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

  3. Taylor expanded in x around 0 0.5

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

  4. Applied egg-rr0.5

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

  5. Final simplification0.5

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

Alternatives

Alternative 1
Error15.2
Cost32896
\[\frac{1}{\sqrt[3]{x} \cdot \left(\sqrt[3]{1 + x} + \sqrt[3]{x}\right) + \sqrt[3]{{\left(1 + x\right)}^{2}}} \]
Alternative 2
Error0.5
Cost32896
\[\begin{array}{l} t_0 := \sqrt[3]{1 + x}\\ \frac{1}{{t_0}^{2} + \sqrt[3]{x} \cdot \left(t_0 + \sqrt[3]{x}\right)} \end{array} \]
Alternative 3
Error29.5
Cost26048
\[{\left({\left(\sqrt[3]{1 + x} - \sqrt[3]{x}\right)}^{3}\right)}^{0.3333333333333333} \]
Alternative 4
Error29.5
Cost13120
\[\sqrt[3]{1 + x} - \sqrt[3]{x} \]
Alternative 5
Error31.3
Cost6592
\[1 - \sqrt[3]{x} \]
Alternative 6
Error61.7
Cost64
\[0 \]
Alternative 7
Error31.9
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022308 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))