Average Error: 1.7 → 0.1
Time: 44.9s
Precision: 64
Internal Precision: 384
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7581267002130027 \cdot 10^{-37}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{if}\;x \le 1.3957929978966932 \cdot 10^{-21}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if x < -1.7581267002130027e-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 simplify0.2

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

    if -1.7581267002130027e-37 < x < 1.3957929978966932e-21

    1. Initial program 2.9

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

    3. Applied associate-*l/0.1

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

    if 1.3957929978966932e-21 < x

    1. Initial program 0.2

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

Runtime

Time bar (total: 44.9s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))