quad2m (problem 3.2.1, negative)

Average Error: 33.6 → 7.6

Time: 4.9s

Precision: binary64

Math

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↓

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Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.0769602548878653e112

   1. Initial program 60.3

      \[\]

   2. Taylor expanded around -inf 2.2

      \[\leadsto \]

   if -1.0769602548878653e112 < b_2 < -3.5889303322404708e-234

   1. Initial program 35.3

      \[\]
   2. Using strategy rm

   3. Applied flip-- 35.4

      \[\leadsto \]

   4. Simplified 16.6

      \[\leadsto \]

   5. Simplified 16.6

      \[\leadsto \]
   6. Using strategy rm

   7. Applied clear-num 16.9

      \[\leadsto \]

   8. Simplified 8.0

      \[\leadsto \]

   if -3.5889303322404708e-234 < b_2 < 1711063662601362.25

   1. Initial program 11.3

      \[\]
   2. Using strategy rm

   3. Applied div-inv 11.4

      \[\leadsto \]

   if 1711063662601362.25 < b_2

   1. Initial program 32.9

      \[\]
   2. Using strategy rm

   3. Applied flip-- 60.0

      \[\leadsto \]

   4. Simplified 59.4

      \[\leadsto \]

   5. Simplified 59.4

      \[\leadsto \]

   6. Taylor expanded around 0 7.5

      \[\leadsto \]

   7. Simplified 7.5

      \[\leadsto \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.6

   \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))