Average Error: 14.7 → 0.1
Time: 1.4m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\frac{-2}{x + 1}}{x - 1}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{\frac{-2}{x + 1}}{x - 1}
double f(double x) {
        double r31704178 = 1.0;
        double r31704179 = x;
        double r31704180 = r31704179 + r31704178;
        double r31704181 = r31704178 / r31704180;
        double r31704182 = r31704179 - r31704178;
        double r31704183 = r31704178 / r31704182;
        double r31704184 = r31704181 - r31704183;
        return r31704184;
}
↓
double f(double x) {
        double r31704185 = -2.0;
        double r31704186 = x;
        double r31704187 = 1.0;
        double r31704188 = r31704186 + r31704187;
        double r31704189 = r31704185 / r31704188;
        double r31704190 = r31704186 - r31704187;
        double r31704191 = r31704189 / r31704190;
        return r31704191;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 14.7

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

  3. Applied flip--29.4

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

  4. Applied associate-/r/29.4

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

  5. Applied flip-+14.7

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

  6. Applied associate-/r/14.7

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

  7. Applied distribute-lft-out--14.1

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

  8. Simplified14.1

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

  9. Simplified0.4

    \[\leadsto \frac{1}{x \cdot x - 1} \cdot \color{blue}{-2}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.4

    \[\leadsto \frac{1}{x \cdot x - \color{blue}{1 \cdot 1}} \cdot -2\]

  12. Applied difference-of-squares0.4

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

  13. Applied associate-/r*0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1}}{x - 1}} \cdot -2\]
  14. Using strategy rm

  15. Applied associate-*l/0.1

    \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot -2}{x - 1}}\]

  16. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{-2}{x + 1}}}{x - 1}\]

  17. Final simplification0.1

    \[\leadsto \frac{\frac{-2}{x + 1}}{x - 1}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))