Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 5.5s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 3.3940743579651296e127

   1. Initial program 0.1

      \[\]
   2. Using strategy rm

   3. Applied associate-/l* 0.1

      \[\leadsto \]

   4. Simplified 0.0

      \[\leadsto \]
   5. Using strategy rm

   6. Applied div-inv 0.0

      \[\leadsto \]

   7. Simplified 0.0

      \[\leadsto \]

   if 3.3940743579651296e127 < k

   1. Initial program 8.2

      \[\]

   2. Taylor expanded around inf 8.2

      \[\leadsto \]

   3. Simplified 0.1

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

4. Final simplification 0.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))