Average Error: 14.5 → 0.0
Time: 2.3m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -409349561137322.562 \lor \neg \left(x \le 170796.26868939324\right):\\ \;\;\;\;1 \cdot \frac{1}{{x}^{3}} - \left(1 \cdot {x}^{\left(-2\right)} + 1 \cdot \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(1 \cdot \frac{x}{\sqrt[3]{1}} - \left(x + 1\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}{\frac{x}{\sqrt[3]{1}}}}{-\left(x + 1\right)}\\ \end{array}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -409349561137322.562 \lor \neg \left(x \le 170796.26868939324\right):\\
\;\;\;\;1 \cdot \frac{1}{{x}^{3}} - \left(1 \cdot {x}^{\left(-2\right)} + 1 \cdot \frac{1}{{x}^{4}}\right)\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (1.0 / ((double) (x + 1.0)))) - ((double) (1.0 / x))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -409349561137322.56) || !(x <= 170796.26868939324))) {
		VAR = ((double) (((double) (1.0 * ((double) (1.0 / ((double) pow(x, 3.0)))))) - ((double) (((double) (1.0 * ((double) pow(x, ((double) -(2.0)))))) + ((double) (1.0 * ((double) (1.0 / ((double) pow(x, 4.0))))))))));
	} else {
		VAR = ((double) (((double) (((double) -(((double) (((double) (1.0 * ((double) (x / ((double) cbrt(1.0)))))) - ((double) (((double) (x + 1.0)) * ((double) (((double) cbrt(1.0)) * ((double) cbrt(1.0)))))))))) / ((double) (x / ((double) cbrt(1.0)))))) / ((double) -(((double) (x + 1.0))))));
	}
	return VAR;
}

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 < -409349561137322.56 or 170796.26868939324 < x

    1. Initial program 29.4

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

    3. Applied add-cube-cbrt29.4

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

    4. Applied associate-/l*29.4

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

    5. Applied frac-2neg29.4

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

    6. Applied frac-sub28.7

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

    7. Simplified28.7

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

    8. Taylor expanded around inf 0.7

      \[\leadsto \color{blue}{1 \cdot \frac{1}{{x}^{3}} - \left(1 \cdot \frac{1}{{x}^{2}} + 1 \cdot \frac{1}{{x}^{4}}\right)}\]
    9. Using strategy rm

    10. Applied pow-flip0.0

      \[\leadsto 1 \cdot \frac{1}{{x}^{3}} - \left(1 \cdot \color{blue}{{x}^{\left(-2\right)}} + 1 \cdot \frac{1}{{x}^{4}}\right)\]

    if -409349561137322.56 < x < 170796.26868939324

    1. Initial program 0.6

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

    3. Applied add-cube-cbrt0.6

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

    4. Applied associate-/l*0.6

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

    5. Applied frac-2neg0.6

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

    6. Applied frac-sub0.0

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

    7. Simplified0.0

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

    9. Applied *-commutative0.0

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

    10. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -409349561137322.562 \lor \neg \left(x \le 170796.26868939324\right):\\ \;\;\;\;1 \cdot \frac{1}{{x}^{3}} - \left(1 \cdot {x}^{\left(-2\right)} + 1 \cdot \frac{1}{{x}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(1 \cdot \frac{x}{\sqrt[3]{1}} - \left(x + 1\right) \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)}{\frac{x}{\sqrt[3]{1}}}}{-\left(x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1 (+ x 1)) (/ 1 x)))