Average Error: 1.6 → 0.4
Time: 8.7m
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.9790678724877975 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 2.176748008187451 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4 - z \cdot x\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -2.9790678724877975e-33 or 2.176748008187451e-129 < x

    1. Initial program 0.6

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

    3. Applied clear-num0.7

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

    4. Applied associate-*l/0.6

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

    5. Simplified0.6

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

    if -2.9790678724877975e-33 < x < 2.176748008187451e-129

    1. Initial program 2.8

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 *-un-lft-identity0.1

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

    5. Applied associate-/r*0.1

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

    6. Applied sub-div0.1

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

    7. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.9790678724877975 \cdot 10^{-33}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 2.176748008187451 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4 - z \cdot x\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{\frac{y}{x}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))