Average Error: 29.9 → 8.7
Time: 5.5s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -78001.84799997171:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \leq 2.9708167353134948 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -78001.84799997171:\\
\;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\

\mathbf{elif}\;x \leq 2.9708167353134948 \cdot 10^{-06}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\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 (<= x -78001.84799997171)
   (-
    (* 0.3333333333333333 (cbrt (/ 1.0 (* x x))))
    (* 0.1111111111111111 (cbrt (/ 1.0 (pow x 5.0)))))
   (if (<= x 2.9708167353134948e-06)
     (-
      (cbrt (* (cbrt (+ x 1.0)) (* (cbrt (+ x 1.0)) (cbrt (+ x 1.0)))))
      (cbrt x))
     (/
      1.0
      (+
       (pow x 0.6666666666666666)
       (* (cbrt (+ 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 <= -78001.84799997171) {
		tmp = (0.3333333333333333 * cbrt(1.0 / (x * x))) - (0.1111111111111111 * cbrt(1.0 / pow(x, 5.0)));
	} else if (x <= 2.9708167353134948e-06) {
		tmp = cbrt(cbrt(x + 1.0) * (cbrt(x + 1.0) * cbrt(x + 1.0))) - cbrt(x);
	} else {
		tmp = 1.0 / (pow(x, 0.6666666666666666) + (cbrt(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 3 regimes

  2. if x < -78001.847999971709

    1. Initial program 60.7

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

    2. Taylor expanded around inf 44.5

      \[\leadsto \color{blue}{0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333}}\]

    3. Simplified31.4

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}}\]

    if -78001.847999971709 < x < 2.97081673531349477e-6

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube_binary64_7960.1

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

    if 2.97081673531349477e-6 < x

    1. Initial program 58.9

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

    3. Applied flip3--_binary64_76458.7

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

    4. Simplified1.0

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

    5. Simplified4.4

      \[\leadsto \frac{1}{\color{blue}{{x}^{0.6666666666666666} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -78001.84799997171:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \leq 2.9708167353134948 \cdot 10^{-06}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right)} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))