Average Error: 1.5 → 0.1
Time: 34.6s
Precision: 64
Internal Precision: 576
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -3.9155860618683845 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + \frac{4}{y}\right|\\ \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 2.3422813505455737 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + \frac{4}{y}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (+ x 4) y) (* (/ x y) z)) < -3.9155860618683845e+35 or 2.3422813505455737e-59 < (- (/ (+ x 4) y) (* (/ x y) z))

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Applied simplify0.1

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

    if -3.9155860618683845e+35 < (- (/ (+ x 4) y) (* (/ x y) z)) < 2.3422813505455737e-59

    1. Initial program 3.9

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

    3. Applied div-inv3.9

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

    4. Applied associate-*l*0.1

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

    5. Applied simplify0.1

      \[\leadsto \left|\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 34.6s)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))