Average Error: 1.5 → 1.5
Time: 14.5s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.5

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]

  2. Taylor expanded around 0 3.5

    \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + 4 \cdot \frac{1}{y}\right) - \frac{x \cdot z}{y}}\right|\]

  3. Simplified1.5

    \[\leadsto \left|\color{blue}{\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{y} \cdot z}\right|\]

  4. Final simplification1.5

    \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - z \cdot \frac{x}{y}\right|\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))