Average Error: 1.5 → 0.2
Time: 10.8s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.214996096676512 \cdot 10^{-49}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \left(\frac{1}{y} \cdot z\right) \cdot x\right|\\ \mathbf{elif}\;x \le 5.4367458134692587 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|(\left(\frac{x}{y}\right) \cdot \left(1 - z\right) + \left(\frac{4}{y}\right))_*\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 3 regimes

  2. if x < -5.214996096676512e-49

    1. Initial program 0.3

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

    2. Taylor expanded around 0 6.7

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

    3. Simplified0.3

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

    5. Applied div-inv0.3

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

    6. Applied associate-*l*0.6

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

    if -5.214996096676512e-49 < x < 5.4367458134692587e-11

    1. Initial program 2.7

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

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

    if 5.4367458134692587e-11 < x

    1. Initial program 0.1

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

    2. Taylor expanded around 0 7.2

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

    3. Simplified0.1

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

    5. Applied add-sqr-sqrt30.5

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

    6. Applied prod-diff30.5

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

    7. Simplified0.1

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

    8. Simplified0.1

      \[\leadsto \left|(\left(\frac{x}{y}\right) \cdot \left(1 - z\right) + \left(\frac{4}{y}\right))_* + \color{blue}{0}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.214996096676512 \cdot 10^{-49}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \left(\frac{1}{y} \cdot z\right) \cdot x\right|\\ \mathbf{elif}\;x \le 5.4367458134692587 \cdot 10^{-11}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|(\left(\frac{x}{y}\right) \cdot \left(1 - z\right) + \left(\frac{4}{y}\right))_*\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))