Average Error: 30.0 → 0.4
Time: 4.5s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -83186.8837971445:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;x \leq 85400.5263612331:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \log \left(e^{\frac{-0.1111111111111111}{x}}\right)\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -83186.8837971445:\\
\;\;\;\;\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{elif}\;x \leq 85400.5263612331:\\
\;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \log \left(e^{\frac{-0.1111111111111111}{x}}\right)\right)\\

\end{array}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (if (<= x -83186.8837971445)
   (+
    (* (/ (cbrt x) x) (+ 0.3333333333333333 (/ -0.1111111111111111 x)))
    (- (cbrt x) (* (cbrt (- x)) (cbrt -1.0))))
   (if (<= x 85400.5263612331)
     (log (exp (- (cbrt (+ x 1.0)) (cbrt x))))
     (+
      (- (cbrt x) (* (cbrt (- x)) (cbrt -1.0)))
      (*
       (/ (cbrt x) x)
       (+ 0.3333333333333333 (log (exp (/ -0.1111111111111111 x)))))))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double tmp;
	if (x <= -83186.8837971445) {
		tmp = ((cbrt(x) / x) * (0.3333333333333333 + (-0.1111111111111111 / x))) + (cbrt(x) - (cbrt(-x) * cbrt(-1.0)));
	} else if (x <= 85400.5263612331) {
		tmp = log(exp(cbrt(x + 1.0) - cbrt(x)));
	} else {
		tmp = (cbrt(x) - (cbrt(-x) * cbrt(-1.0))) + ((cbrt(x) / x) * (0.3333333333333333 + log(exp(-0.1111111111111111 / 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 3 regimes

  2. if x < -83186.8837971445028

    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 -83186.8837971445028 < x < 85400.5263612331037

    1. Initial program 0.2

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

    3. Applied add-log-exp_binary64_4580.2

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

    4. Applied add-log-exp_binary64_4580.2

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

    5. Applied diff-log_binary64_5110.2

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

    6. Simplified0.2

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

    if 85400.5263612331037 < 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

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

    5. Applied add-log-exp_binary64_4580.7

      \[\leadsto \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \color{blue}{\log \left(e^{\frac{-0.1111111111111111}{x}}\right)}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -83186.8837971445:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \frac{-0.1111111111111111}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right)\\ \mathbf{elif}\;x \leq 85400.5263612331:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \sqrt[3]{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-x} \cdot \sqrt[3]{-1}\right) + \frac{\sqrt[3]{x}}{x} \cdot \left(0.3333333333333333 + \log \left(e^{\frac{-0.1111111111111111}{x}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020354 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))