Average Error: 14.7 → 0.1
Time: 11.1s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x - 1}\]
↓
\[\left(2 \cdot \frac{1}{x - 1}\right) \cdot \frac{-1}{x + 1}\]
\frac{1}{x + 1} - \frac{1}{x - 1}
↓
\left(2 \cdot \frac{1}{x - 1}\right) \cdot \frac{-1}{x + 1}
double f(double x) {
        double r4640020 = 1.0;
        double r4640021 = x;
        double r4640022 = r4640021 + r4640020;
        double r4640023 = r4640020 / r4640022;
        double r4640024 = r4640021 - r4640020;
        double r4640025 = r4640020 / r4640024;
        double r4640026 = r4640023 - r4640025;
        return r4640026;
}
↓
double f(double x) {
        double r4640027 = 2.0;
        double r4640028 = 1.0;
        double r4640029 = x;
        double r4640030 = r4640029 - r4640028;
        double r4640031 = r4640028 / r4640030;
        double r4640032 = r4640027 * r4640031;
        double r4640033 = -1.0;
        double r4640034 = r4640029 + r4640028;
        double r4640035 = r4640033 / r4640034;
        double r4640036 = r4640032 * r4640035;
        return r4640036;
}

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--28.7

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

  4. Applied associate-/r/28.7

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

  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. Taylor expanded around 0 0.3

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

  10. Applied difference-of-squares0.3

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

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

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

  12. Applied times-frac0.1

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

  13. Applied associate-*l*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Asymptote A"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))