Jmat.Real.lambertw, newton loop step

Average Error: 13.5 → 0.9

Time: 3.9s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus wj

Bits error versus x

Target

Original	13.5
Target	13.0
Herbie	0.9

\[\]

Derivation

1. Split input into 2 regimes

2. if wj < 4.47640317399307113e-9

   1. Initial program 13.2

      \[\]

   2. Simplified 13.2

      \[\leadsto \]

   3. Taylor expanded around 0 0.8

      \[\leadsto \]

   4. Simplified 0.9

      \[\leadsto \]

   if 4.47640317399307113e-9 < wj

   1. Initial program 24.6

      \[\]

   2. Simplified 3.1

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 3.1

      \[\leadsto \]

   5. Simplified 3.1

      \[\leadsto \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (wj x)
  :name "Jmat.Real.lambertw, newton loop step"
  :precision binary64

  :herbie-target
  (- wj (- (/ wj (+ wj 1.0)) (/ x (+ (exp wj) (* wj (exp wj))))))

  (- wj (/ (- (* wj (exp wj)) x) (+ (exp wj) (* wj (exp wj))))))