Average Error: 29.7 → 0.2
Time: 5.8s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\frac{\frac{-\left(3 \cdot x + 1\right)}{x + 1}}{x - 1}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\frac{\frac{-\left(3 \cdot x + 1\right)}{x + 1}}{x - 1}
double f(double x) {
        double r86611 = x;
        double r86612 = 1.0;
        double r86613 = r86611 + r86612;
        double r86614 = r86611 / r86613;
        double r86615 = r86611 - r86612;
        double r86616 = r86613 / r86615;
        double r86617 = r86614 - r86616;
        return r86617;
}
↓
double f(double x) {
        double r86618 = 3.0;
        double r86619 = x;
        double r86620 = r86618 * r86619;
        double r86621 = 1.0;
        double r86622 = r86620 + r86621;
        double r86623 = -r86622;
        double r86624 = r86619 + r86621;
        double r86625 = r86623 / r86624;
        double r86626 = r86619 - r86621;
        double r86627 = r86625 / r86626;
        return r86627;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 29.7

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

  3. Applied *-un-lft-identity29.7

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

  4. Applied add-cube-cbrt30.1

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

  5. Applied times-frac30.2

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

  6. Simplified30.2

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

  8. Applied associate-*r/30.1

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

  9. Applied frac-sub30.8

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

  10. Simplified30.8

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

  11. Taylor expanded around 0 14.9

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

  13. Applied associate-/r*0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019308 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))