Average Error: 29.4 → 0.4
Time: 2.8s
Precision: binary64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -58198.39330764905 \lor \neg \left(x \leq 58716.682115847965\right):\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{-0.1111111111111111}{x} + 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -58198.39330764905 \lor \neg \left(x \leq 58716.682115847965\right):\\
\;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{-0.1111111111111111}{x} + 0.3333333333333333\right)}{x}\\

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

\end{array}
(FPCore (x) :precision binary64 (- (cbrt (+ x 1.0)) (cbrt x)))
↓
(FPCore (x)
 :precision binary64
 (if (or (<= x -58198.39330764905) (not (<= x 58716.682115847965)))
   (/ (* (cbrt x) (+ (/ -0.1111111111111111 x) 0.3333333333333333)) x)
   (- (/ (cbrt (+ 1.0 (pow x 3.0))) (cbrt (+ (* x x) (- 1.0 x)))) (cbrt x))))
double code(double x) {
	return cbrt(x + 1.0) - cbrt(x);
}
↓
double code(double x) {
	double tmp;
	if ((x <= -58198.39330764905) || !(x <= 58716.682115847965)) {
		tmp = (cbrt(x) * ((-0.1111111111111111 / x) + 0.3333333333333333)) / x;
	} else {
		tmp = (cbrt(1.0 + pow(x, 3.0)) / cbrt((x * x) + (1.0 - x))) - 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 < -58198.39330764905 or 58716.6821158479652 < x

    1. Initial program 60.4

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

    2. Taylor expanded around -inf 64.0

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

    3. Simplified0.7

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

    5. Applied associate-*l/_binary640.7

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

    6. Simplified0.7

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

    if -58198.39330764905 < x < 58716.6821158479652

    1. Initial program 0.2

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

    3. Applied flip3-+_binary640.2

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

    4. Applied cbrt-div_binary640.2

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

    5. Simplified0.2

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

    6. Simplified0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -58198.39330764905 \lor \neg \left(x \leq 58716.682115847965\right):\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \left(\frac{-0.1111111111111111}{x} + 0.3333333333333333\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 + {x}^{3}}}{\sqrt[3]{x \cdot x + \left(1 - x\right)}} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1.0)) (cbrt x)))