Average Error: 30.8 → 0.4
Time: 4.4s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -71136.05163775448 \lor \neg \left(x \leq 76227.4301677554\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -71136.05163775448 \lor \neg \left(x \leq 76227.4301677554\right):\\
\;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -71136.05163775448) (not (<= x 76227.4301677554)))
   (+
    (* (/ (cbrt x) x) (+ 0.3333333333333333 (/ -0.1111111111111111 x)))
    (- (cbrt x) (* (cbrt (- x)) (cbrt -1.0))))
   (-
    (cbrt (+ x 1.0))
    (* (cbrt (cbrt x)) (* (cbrt (cbrt x)) (cbrt (cbrt x)))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -71136.05163775448) || !(x <= 76227.4301677554)) {
		tmp = ((cbrt(x) / x) * (0.3333333333333333 + (-0.1111111111111111 / x))) + (cbrt(x) - (cbrt(-x) * cbrt(-1.0)));
	} else {
		tmp = cbrt(x + 1.0) - (cbrt(cbrt(x)) * (cbrt(cbrt(x)) * cbrt(cbrt(x))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -71136.051637754485 or 76227.430167755403 < x

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 64.0

      \[\leadsto \color{blue}{\left(e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + 0.3333333333333333 \cdot \frac{e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \left({\left(-1 \cdot x\right)}^{0.3333333333333333} \cdot \sqrt[3]{-1} + 0.1111111111111111 \cdot \frac{e^{0.3333333333333333 \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}\]

    3. Simplified0.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)}\]

    if -71136.051637754485 < x < 76227.430167755403

    1. Initial program 0.1

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary640.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -71136.05163775448 \lor \neg \left(x \leq 76227.4301677554\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020260 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))