Average Error: 1.5 → 1.5
Time: 22.4s
Precision: 64
Internal Precision: 320
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{\frac{x}{y}}{\frac{1}{z}}\right|\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.5

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

  2. Taylor expanded around inf 3.6

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

  3. Simplified2.9

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

  5. Applied div-inv3.0

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

  6. Applied associate-/r*1.5

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

  7. Final simplification1.5

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

Runtime

Time bar (total: 22.4s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))