jeff quadratic root 2

Average Error: 19.2 → 6.5

Time: 6.6s

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 < -6.0329111911809092e90

   1. Initial program 45.1

      \[\]

   2. Taylor expanded around -inf 11.0

      \[\leadsto \]

   3. Simplified 5.2

      \[\leadsto \]

   4. Taylor expanded around 0 5.0

      \[\leadsto \]

   5. Simplified 5.0

      \[\leadsto \]
   6. Using strategy rm

   7. Applied div-inv 5.0

      \[\leadsto \]

   8. Simplified 5.0

      \[\leadsto \]

   if -6.0329111911809092e90 < b < -2.6862915384868101e-246 or 1.35418184362108912e101 < b

   1. Initial program 18.1

      \[\]

   2. Taylor expanded around inf 6.9

      \[\leadsto \]

   3. Simplified 5.1

      \[\leadsto \]

   if -2.6862915384868101e-246 < b < 1.35418184362108912e101

   1. Initial program 9.0

      \[\]
   2. Using strategy rm

   3. Applied flip-+ 9.0

      \[\leadsto \]

   4. Simplified 9.1

      \[\leadsto \]

   5. Simplified 9.0

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

4. Final simplification 6.5

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (neg b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))