Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Average Error: 12.0 → 0.1

Time: 6.7s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

↓

\[x - \frac{-2}{\frac{t}{z} - \frac{2}{\frac{y}{z}}}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	12.0
Target	0.1
Herbie	0.1

\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}}\]

Derivation

1. Initial program 12.0

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\]

2. Simplified 0.1

   \[\leadsto \color{blue}{x - \frac{2}{\frac{z \cdot 2}{y} - \frac{t}{z}}}\]
3. Using strategy rm

4. Applied frac-2neg 0.1

   \[\leadsto x - \color{blue}{\frac{-2}{-\left(\frac{z \cdot 2}{y} - \frac{t}{z}\right)}}\]

5. Simplified 0.1

   \[\leadsto x - \frac{-2}{\color{blue}{\frac{t}{z} - \frac{2}{\frac{y}{z}}}}\]

6. Final simplification 0.1

   \[\leadsto x - \frac{-2}{\frac{t}{z} - \frac{2}{\frac{y}{z}}}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))