Average Error: 13.6 → 0.5
Time: 28.3s
Precision: 64
\[\frac{1}{x + 1} - \frac{1}{x}\]
↓
\[\frac{-1}{\mathsf{fma}\left(x, x, x\right)}\]
\frac{1}{x + 1} - \frac{1}{x}
↓
\frac{-1}{\mathsf{fma}\left(x, x, x\right)}
double f(double x) {
        double r1613585 = 1.0;
        double r1613586 = x;
        double r1613587 = r1613586 + r1613585;
        double r1613588 = r1613585 / r1613587;
        double r1613589 = r1613585 / r1613586;
        double r1613590 = r1613588 - r1613589;
        return r1613590;
}
↓
double f(double x) {
        double r1613591 = -1.0;
        double r1613592 = x;
        double r1613593 = fma(r1613592, r1613592, r1613592);
        double r1613594 = r1613591 / r1613593;
        return r1613594;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.6

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

  3. Applied frac-sub13.0

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

  4. Simplified13.0

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

  5. Simplified13.0

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

  7. Applied *-un-lft-identity13.0

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

  8. Applied associate-/r*13.0

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

  9. Simplified0.5

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

  10. Final simplification0.5

    \[\leadsto \frac{-1}{\mathsf{fma}\left(x, x, x\right)}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  (- (/ 1 (+ x 1)) (/ 1 x)))