Quadratic roots, full range

Average Error: 34.2 → 10.3

Time: 5.3s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.6233005911807882e76

   1. Initial program 41.8

      \[\]

   2. Simplified 41.8

      \[\leadsto \]

   3. Taylor expanded around -inf 4.5

      \[\leadsto \]

   4. Simplified 4.5

      \[\leadsto \]

   if -1.6233005911807882e76 < b < 4.2628474888354611e-38

   1. Initial program 14.6

      \[\]

   2. Simplified 14.6

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-sub 14.6

      \[\leadsto \]

   if 4.2628474888354611e-38 < b

   1. Initial program 55.1

      \[\]

   2. Simplified 55.0

      \[\leadsto \]

   3. Taylor expanded around inf 7.6

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

4. Final simplification 10.3

   \[\leadsto \]

Reproduce

herbie shell --seed 2020190 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))