Average Error: 1.6 → 0.1
Time: 35.8s
Precision: 64
Internal Precision: 384
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.252315155049909 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \le 328395115868051.0:\\ \;\;\;\;\left|\frac{1}{\frac{y}{\left(x + 4\right) - x \cdot z}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if x < -6.252315155049909e-37

    1. Initial program 0.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Applied simplify0.1

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

    if -6.252315155049909e-37 < x < 328395115868051.0

    1. Initial program 2.6

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    6. Applied clear-num0.1

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

    if 328395115868051.0 < x

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 35.8s)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))