Average Error: 1.4 → 0.2
Time: 14.3s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.276086576151565 \cdot 10^{-06}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(z \cdot \frac{1}{y}\right)\right|\\ \mathbf{elif}\;x \le 2.3141640756322242 \cdot 10^{-07}:\\ \;\;\;\;\left|\frac{\left(y \cdot \left(\frac{4}{y} + \frac{x}{y}\right) - z \cdot x\right) \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(z \cdot \frac{1}{y}\right)\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -8.276086576151565e-06 or 2.3141640756322242e-07 < x

    1. Initial program 0.1

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

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

    if -8.276086576151565e-06 < x < 2.3141640756322242e-07

    1. Initial program 2.3

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

    3. Applied div-inv2.3

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

    4. Applied associate-*l*5.6

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

    5. Taylor expanded around 0 0.1

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

    6. Simplified2.3

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

    8. Applied associate-*l/0.1

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

    9. Applied flip-+24.5

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

    10. Applied frac-sub24.6

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

    11. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.276086576151565 \cdot 10^{-06}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(z \cdot \frac{1}{y}\right)\right|\\ \mathbf{elif}\;x \le 2.3141640756322242 \cdot 10^{-07}:\\ \;\;\;\;\left|\frac{\left(y \cdot \left(\frac{4}{y} + \frac{x}{y}\right) - z \cdot x\right) \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \left(z \cdot \frac{1}{y}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))