Average Error: 15.2 → 0.1
Time: 1.4m
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\frac{\frac{-2}{x + 1}}{x - 1}\]
double f(double x) {
        double r18982098 = 1.0;
        double r18982099 = x;
        double r18982100 = r18982099 + r18982098;
        double r18982101 = r18982098 / r18982100;
        double r18982102 = r18982099 - r18982098;
        double r18982103 = r18982098 / r18982102;
        double r18982104 = r18982101 - r18982103;
        return r18982104;
}
↓
double f(double x) {
        double r18982105 = -2.0;
        double r18982106 = x;
        double r18982107 = 1.0;
        double r18982108 = r18982106 + r18982107;
        double r18982109 = r18982105 / r18982108;
        double r18982110 = r18982106 - r18982107;
        double r18982111 = r18982109 / r18982110;
        return r18982111;
}


\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\frac{\frac{-2}{x + 1}}{x - 1}

Error

Bits error versus x

Derivation

  1. Initial program 15.2

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

  3. Applied flip--29.2

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

  4. Applied associate-/r/29.2

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

  5. Applied flip-+15.3

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

  6. Applied associate-/r/15.2

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

  7. Applied distribute-lft-out--14.6

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

  8. Simplified14.6

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

  9. Simplified0.3

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

  11. Applied *-un-lft-identity0.3

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

  12. Applied difference-of-squares0.3

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

  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 2019101 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1 (+ x 1)) (/ 1 (- x 1))))