Average Error: 14.6 → 0.1
Time: 11.3s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{\frac{-1}{1 + x}}{x}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{\frac{-1}{1 + x}}{x}
double f(double x) {
        double r897074 = 1.0;
        double r897075 = x;
        double r897076 = r897075 + r897074;
        double r897077 = r897074 / r897076;
        double r897078 = r897074 / r897075;
        double r897079 = r897077 - r897078;
        return r897079;
}
↓
double f(double x) {
        double r897080 = -1.0;
        double r897081 = 1.0;
        double r897082 = x;
        double r897083 = r897081 + r897082;
        double r897084 = r897080 / r897083;
        double r897085 = r897084 / r897082;
        return r897085;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.6

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

  3. Applied frac-sub14.0

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

  4. Simplified14.0

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

  5. Simplified14.0

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

  7. Applied associate-/r*14.0

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

  8. Taylor expanded around -inf 0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))