Quadratic roots, full range

Average Error: 34.9 → 7.5

Time: 5.6s

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 < -1.52263982968363542e79

   1. Initial program 42.4

      \[\]

   2. Taylor expanded around -inf 5.2

      \[\leadsto \]

   3. Simplified 5.2

      \[\leadsto \]

   if -1.52263982968363542e79 < b < -5.17271316252318646e-305

   1. Initial program 9.6

      \[\]

   if -5.17271316252318646e-305 < b < 5.954633135651186e40

   1. Initial program 30.0

      \[\]
   2. Using strategy rm

   3. Applied flip-+ 30.0

      \[\leadsto \]

   4. Simplified 18.3

      \[\leadsto \]

   5. Simplified 18.3

      \[\leadsto \]
   6. Using strategy rm

   7. Applied *-un-lft-identity 18.3

      \[\leadsto \]

   8. Applied times-frac 18.3

      \[\leadsto \]

   9. Applied times-frac 18.3

      \[\leadsto \]

   10. Simplified 18.3

       \[\leadsto \]

   11. Simplified 10.7

       \[\leadsto \]

   if 5.954633135651186e40 < b

   1. Initial program 57.5

      \[\]

   2. Taylor expanded around inf 4.3

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

4. Final simplification 7.5

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))