Quadratic roots, full range

Average Error: 34.5 → 6.8

Time: 5.8s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -4.76974310350345068e116

   1. Initial program 51.1

      \[\]

   2. Simplified 51.1

      \[\leadsto \]

   3. Taylor expanded around -inf 3.2

      \[\leadsto \]

   4. Simplified 3.2

      \[\leadsto \]

   if -4.76974310350345068e116 < b < -2.6312571937593078e-192

   1. Initial program 7.0

      \[\]

   2. Simplified 7.0

      \[\leadsto \]
   3. Using strategy rm

   4. Applied *-un-lft-identity 7.0

      \[\leadsto \]

   5. Applied times-frac 7.1

      \[\leadsto \]

   if -2.6312571937593078e-192 < b < 4.26893222550519992e95

   1. Initial program 29.8

      \[\]

   2. Simplified 29.8

      \[\leadsto \]
   3. Using strategy rm

   4. Applied clear-num 29.8

      \[\leadsto \]

   5. Simplified 29.8

      \[\leadsto \]
   6. Using strategy rm

   7. Applied flip-- 30.0

      \[\leadsto \]

   8. Simplified 17.0

      \[\leadsto \]

   9. Simplified 17.0

      \[\leadsto \]
   10. Using strategy rm

   11. Applied associate-/r/ 16.4

       \[\leadsto \]

   12. Simplified 10.4

       \[\leadsto \]

   if 4.26893222550519992e95 < b

   1. Initial program 59.2

      \[\]

   2. Simplified 59.2

      \[\leadsto \]

   3. Taylor expanded around inf 3.0

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

4. Final simplification 6.8

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))