Quadratic roots, narrow range

Average Error: 28.4 → 16.2

Time: 7.6s

Precision: binary64

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Math

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↓

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Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 486.3784181750889

   1. Initial program 16.2

      \[\]

   2. Simplified 16.2

      \[\leadsto \]
   3. Using strategy rm

   4. Applied flip-- 16.2

      \[\leadsto \]

   5. Simplified 15.2

      \[\leadsto \]

   6. Simplified 15.2

      \[\leadsto \]

   if 486.3784181750889 < b

   1. Initial program 35.6

      \[\]

   2. Simplified 35.6

      \[\leadsto \]

   3. Taylor expanded around inf 16.8

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

4. Final simplification 16.2

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))