Quadratic roots, medium range

Average Error: 44.3 → 10.8

Time: 5.7s

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 < 1.05967144796170832e-4

   1. Initial program 18.8

      \[\]

   2. Simplified 18.8

      \[\leadsto \]
   3. Using strategy rm

   4. Applied flip-- 18.8

      \[\leadsto \]

   5. Simplified 17.8

      \[\leadsto \]

   6. Simplified 17.8

      \[\leadsto \]

   if 1.05967144796170832e-4 < b

   1. Initial program 46.1

      \[\]

   2. Simplified 46.1

      \[\leadsto \]

   3. Taylor expanded around inf 10.3

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

4. Final simplification 10.8

   \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))