Average Error: 29.4 → 0.4
Time: 3.7s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -125571.14809576281 \lor \neg \left(x \leq 77290.66097568852\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -125571.14809576281 \lor \neg \left(x \leq 77290.66097568852\right):\\
\;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}}\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -125571.14809576281) (not (<= x 77290.66097568852)))
   (+
    (* (/ (cbrt x) x) (+ 0.3333333333333333 (/ -0.1111111111111111 x)))
    (- (cbrt x) (cbrt x)))
   (log (exp (- (/ (cbrt (- (* x x) 1.0)) (cbrt (- x 1.0))) (cbrt x))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -125571.14809576281) || !(x <= 77290.66097568852)) {
		tmp = ((cbrt(x) / x) * (0.3333333333333333 + (-0.1111111111111111 / x))) + (cbrt(x) - cbrt(x));
	} else {
		tmp = log(exp((cbrt((x * x) - 1.0) / cbrt(x - 1.0)) - 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 < -125571.148095762808 or 77290.660975688515 < x

    1. Initial program 60.3

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

    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

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

    if -125571.148095762808 < x < 77290.660975688515

    1. Initial program 0.2

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

    3. Applied flip-+_binary64_3880.2

      \[\leadsto \sqrt[3]{\color{blue}{\frac{x \cdot x - 1 \cdot 1}{x - 1}}} - \sqrt[3]{x}\]

    4. Applied cbrt-div_binary64_4430.2

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}}} - \sqrt[3]{x}\]

    5. Simplified0.2

      \[\leadsto \frac{\color{blue}{\sqrt[3]{x \cdot x - 1}}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\]
    6. Using strategy rm

    7. Applied add-log-exp_binary64_4500.2

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

    8. Applied add-log-exp_binary64_4500.2

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

    9. Applied diff-log_binary64_5030.2

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

    10. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125571.14809576281 \lor \neg \left(x \leq 77290.66097568852\right):\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\sqrt[3]{x \cdot x - 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))