Average Error: 30.2 → 9.6
Time: 2.4s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -4083.0155412597046:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \leq 2.485660549684957 \cdot 10^{-09}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{6}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -4083.0155412597046:\\
\;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\

\mathbf{elif}\;x \leq 2.485660549684957 \cdot 10^{-09}:\\
\;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{6}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \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 -4083.0155412597046)
   (-
    (+
     (* 0.3333333333333333 (cbrt (/ 1.0 (* x x))))
     (* 0.06172839506172839 (cbrt (/ 1.0 (pow x 8.0)))))
    (* 0.1111111111111111 (cbrt (/ 1.0 (pow x 5.0)))))
   (if (<= x 2.485660549684957e-09)
     (-
      (* (cbrt (cbrt (pow (cbrt (+ x 1.0)) 6.0))) (cbrt (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 ((double) (((double) cbrt(((double) (x + 1.0)))) - ((double) cbrt(x))));
}
↓
double code(double x) {
	double tmp;
	if ((x <= -4083.0155412597046)) {
		tmp = ((double) (((double) (((double) (0.3333333333333333 * ((double) cbrt((1.0 / ((double) (x * x))))))) + ((double) (0.06172839506172839 * ((double) cbrt((1.0 / ((double) pow(x, 8.0))))))))) - ((double) (0.1111111111111111 * ((double) cbrt((1.0 / ((double) pow(x, 5.0)))))))));
	} else {
		double tmp_1;
		if ((x <= 2.485660549684957e-09)) {
			tmp_1 = ((double) (((double) (((double) cbrt(((double) cbrt(((double) pow(((double) cbrt(((double) (x + 1.0)))), 6.0)))))) * ((double) cbrt(((double) cbrt(((double) (x + 1.0)))))))) - ((double) cbrt(x))));
		} else {
			tmp_1 = (1.0 / ((double) (((double) pow(x, 0.6666666666666666)) + ((double) (((double) cbrt(((double) (x + 1.0)))) * ((double) (((double) cbrt(((double) (x + 1.0)))) + ((double) cbrt(x)))))))));
		}
		tmp = tmp_1;
	}
	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 < -4083.01554125970461

    1. Initial program 60.2

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

    2. Taylor expanded around inf 45.1

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{0.3333333333333333} + 0.06172839506172839 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{0.3333333333333333}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{0.3333333333333333}}\]

    3. Simplified32.4

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}}\]

    if -4083.01554125970461 < x < 2.48566054968495714e-9

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt_binary640.1

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

    4. Applied cbrt-prod_binary640.1

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

    6. Applied cbrt-unprod_binary640.1

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

    7. Simplified0.1

      \[\leadsto \sqrt[3]{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{x + 1}\right)}^{6}}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\]

    if 2.48566054968495714e-9 < x

    1. Initial program 57.8

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

    3. Applied flip3--_binary6457.6

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

    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

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4083.0155412597046:\\ \;\;\;\;\left(0.3333333333333333 \cdot \sqrt[3]{\frac{1}{x \cdot x}} + 0.06172839506172839 \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \leq 2.485660549684957 \cdot 10^{-09}:\\ \;\;\;\;\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{x + 1}\right)}^{6}}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{0.6666666666666666} + \sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))