Average Error: 34.5 → 7.0
Time: 5.8s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -9.54347509538870147e64

    1. Initial program 39.6

      \[\]

    2. Taylor expanded around -inf 5.0

      \[\leadsto \]

    3. Simplified5.0

      \[\leadsto \]

    if -9.54347509538870147e64 < b < -1.1275433833610337e-179

    1. Initial program 8.2

      \[\]
    2. Using strategy rm

    3. Applied associate-/r*8.2

      \[\leadsto \]

    4. Simplified8.1

      \[\leadsto \]

    if -1.1275433833610337e-179 < b < 6.5883209731476263e95

    1. Initial program 29.1

      \[\]
    2. Using strategy rm

    3. Applied flip-+29.3

      \[\leadsto \]

    4. Simplified17.0

      \[\leadsto \]

    5. Simplified16.9

      \[\leadsto \]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.9

      \[\leadsto \]

    8. Applied times-frac16.9

      \[\leadsto \]

    9. Applied times-frac16.8

      \[\leadsto \]

    10. Simplified16.8

      \[\leadsto \]

    11. Simplified10.3

      \[\leadsto \]

    if 6.5883209731476263e95 < b

    1. Initial program 59.2

      \[\]

    2. Taylor expanded around inf 2.7

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

  4. Final simplification7.0

    \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))