Average Error: 23.6 → 23.7
Time: 8.3s
Precision: binary64
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
↓
\[\frac{x \cdot y + z \cdot \left(t - a\right)}{\left(y + b \cdot z\right) + z \cdot \left(-y\right)}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original23.6
Target17.9
Herbie23.7
\[\frac{z \cdot t + y \cdot x}{y + z \cdot \left(b - y\right)} - \frac{a}{\left(b - y\right) + \frac{y}{z}}\]

Derivation

  1. Initial program 23.6

    \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\]
  2. Using strategy rm

  3. Applied sub-neg23.6

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \color{blue}{\left(b + \left(-y\right)\right)}}\]

  4. Applied distribute-lft-in23.7

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{y + \color{blue}{\left(z \cdot b + z \cdot \left(-y\right)\right)}}\]

  5. Applied associate-+r+23.7

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{\color{blue}{\left(y + z \cdot b\right) + z \cdot \left(-y\right)}}\]

  6. Simplified23.7

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{\color{blue}{\left(y + b \cdot z\right)} + z \cdot \left(-y\right)}\]

  7. Final simplification23.7

    \[\leadsto \frac{x \cdot y + z \cdot \left(t - a\right)}{\left(y + b \cdot z\right) + z \cdot \left(-y\right)}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :herbie-target
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))