Average Error: 14.8 → 0.1
Time: 13.9s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[-\frac{x - 1}{x} \cdot \frac{\frac{1}{x + 1}}{x - 1}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
-\frac{x - 1}{x} \cdot \frac{\frac{1}{x + 1}}{x - 1}
double f(double x) {
        double r2742584 = 1.0;
        double r2742585 = x;
        double r2742586 = r2742585 + r2742584;
        double r2742587 = r2742584 / r2742586;
        double r2742588 = r2742584 / r2742585;
        double r2742589 = r2742587 - r2742588;
        return r2742589;
}
↓
double f(double x) {
        double r2742590 = x;
        double r2742591 = 1.0;
        double r2742592 = r2742590 - r2742591;
        double r2742593 = r2742592 / r2742590;
        double r2742594 = r2742590 + r2742591;
        double r2742595 = r2742591 / r2742594;
        double r2742596 = r2742595 / r2742592;
        double r2742597 = r2742593 * r2742596;
        double r2742598 = -r2742597;
        return r2742598;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 14.8

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

  3. Applied frac-sub14.2

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

  4. Taylor expanded around 0 0.4

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

  6. Applied associate-/r*0.1

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

  8. Applied *-un-lft-identity0.1

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

  9. Applied flip-+0.5

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

  10. Applied associate-/r/0.5

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

  11. Applied times-frac0.4

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))