Average Error: 1.4 → 0.3
Time: 15.8s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.197302185245373 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.904548388227548 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(\left(\frac{x}{y} + \frac{4}{y}\right) \cdot y - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -5.197302185245373e+26 or 2.904548388227548e-69 < x

    1. Initial program 0.3

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

    3. Applied div-inv0.4

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

    if -5.197302185245373e+26 < x < 2.904548388227548e-69

    1. Initial program 2.3

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified5.7

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

    5. Applied associate-*r/0.1

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

    6. Applied flip-+24.7

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

    7. Applied frac-sub24.8

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

    8. 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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.197302185245373 \cdot 10^{+26}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.904548388227548 \cdot 10^{-69}:\\ \;\;\;\;\left|\frac{\left(\frac{x}{y} - \frac{4}{y}\right) \cdot \left(\left(\frac{x}{y} + \frac{4}{y}\right) \cdot y - x \cdot z\right)}{y \cdot \left(\frac{x}{y} - \frac{4}{y}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(4 + x\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

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