Average Error: 1.5 → 2.3
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{4 + x}{y} - \frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot x}{y} \cdot \sqrt[3]{z}\right|\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.5

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

  3. Applied add-cube-cbrt1.8

    \[\leadsto \left|\frac{x + 4}{y} - \frac{x}{y} \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right|\]

  4. Applied associate-*r*1.8

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\frac{x}{y} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}}\right|\]
  5. Using strategy rm

  6. Applied associate-*l/2.3

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)}{y}} \cdot \sqrt[3]{z}\right|\]

  7. Final simplification2.3

    \[\leadsto \left|\frac{4 + x}{y} - \frac{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot x}{y} \cdot \sqrt[3]{z}\right|\]

Reproduce

herbie shell --seed 2019094 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))