Average Error: 29.0 → 0.8
Time: 15.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\frac{-\sqrt[3]{3 \cdot x + 1} \cdot \sqrt[3]{3 \cdot x + 1}}{x - 1} \cdot \frac{\sqrt[3]{3 \cdot x + 1}}{1 + x}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\frac{-\sqrt[3]{3 \cdot x + 1} \cdot \sqrt[3]{3 \cdot x + 1}}{x - 1} \cdot \frac{\sqrt[3]{3 \cdot x + 1}}{1 + x}
double f(double x) {
        double r93288 = x;
        double r93289 = 1.0;
        double r93290 = r93288 + r93289;
        double r93291 = r93288 / r93290;
        double r93292 = r93288 - r93289;
        double r93293 = r93290 / r93292;
        double r93294 = r93291 - r93293;
        return r93294;
}
↓
double f(double x) {
        double r93295 = 3.0;
        double r93296 = x;
        double r93297 = r93295 * r93296;
        double r93298 = 1.0;
        double r93299 = r93297 + r93298;
        double r93300 = cbrt(r93299);
        double r93301 = r93300 * r93300;
        double r93302 = -r93301;
        double r93303 = r93296 - r93298;
        double r93304 = r93302 / r93303;
        double r93305 = r93298 + r93296;
        double r93306 = r93300 / r93305;
        double r93307 = r93304 * r93306;
        return r93307;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.0

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

  3. Applied frac-sub30.1

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

  4. Simplified30.1

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

  5. Taylor expanded around 0 14.9

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

  7. Applied add-cube-cbrt15.1

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

  8. Applied distribute-lft-neg-in15.1

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

  9. Applied times-frac0.8

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

  10. Final simplification0.8

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

Reproduce

herbie shell --seed 2019306 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))