jeff quadratic root 2

Average Error: 19.5 → 6.7

Time: 6.5s

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 < -3.27738173899066951e67

   1. Initial program 41.4

      \[\]

   2. Taylor expanded around -inf 10.4

      \[\leadsto \]

   3. Simplified 4.4

      \[\leadsto \]

   4. Taylor expanded around 0 4.3

      \[\leadsto \]

   5. Simplified 4.3

      \[\leadsto \]

   if -3.27738173899066951e67 < b < 4.4305067213711337e95

   1. Initial program 9.0

      \[\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 9.2

      \[\leadsto \]

   4. Applied sqrt-prod 9.2

      \[\leadsto \]

   5. Simplified 9.2

      \[\leadsto \]

   6. Simplified 9.2

      \[\leadsto \]

   if 4.4305067213711337e95 < b

   1. Initial program 28.5

      \[\]

   2. Taylor expanded around inf 6.0

      \[\leadsto \]

   3. Simplified 2.3

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

4. Final simplification 6.7

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(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))))