Average Error: 34.1 → 7.3
Time: 5.7s
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 < -6.0941843680176984e51

    1. Initial program 38.6

      \[\]

    2. Taylor expanded around -inf 6.1

      \[\leadsto \]

    3. Simplified6.1

      \[\leadsto \]

    if -6.0941843680176984e51 < b < -7.40818469346522838e-233

    1. Initial program 9.1

      \[\]
    2. Using strategy rm

    3. Applied *-un-lft-identity9.1

      \[\leadsto \]

    4. Applied times-frac9.3

      \[\leadsto \]

    5. Simplified9.2

      \[\leadsto \]

    if -7.40818469346522838e-233 < b < 4.7963432278148011e75

    1. Initial program 28.6

      \[\]
    2. Using strategy rm

    3. Applied flip-+28.7

      \[\leadsto \]

    4. Simplified16.2

      \[\leadsto \]

    5. Simplified16.1

      \[\leadsto \]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.1

      \[\leadsto \]

    8. Applied times-frac16.1

      \[\leadsto \]

    9. Applied times-frac16.0

      \[\leadsto \]

    10. Simplified16.0

      \[\leadsto \]

    11. Simplified9.8

      \[\leadsto \]

    if 4.7963432278148011e75 < b

    1. Initial program 58.4

      \[\]

    2. Taylor expanded around inf 3.4

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

  4. Final simplification7.3

    \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))