Average Error: 34.9 → 6.9
Time: 4.5s
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 < -1.19922642763792684e154

    1. Initial program 63.9

      \[\]

    2. Taylor expanded around -inf 11.5

      \[\leadsto \]

    3. Simplified2.4

      \[\leadsto \]

    if -1.19922642763792684e154 < b < -3.7072738864736419e-240

    1. Initial program 9.0

      \[\]
    2. Using strategy rm

    3. Applied clear-num9.0

      \[\leadsto \]

    4. Simplified9.0

      \[\leadsto \]

    if -3.7072738864736419e-240 < b < 2.35376843954459236e133

    1. Initial program 32.6

      \[\]
    2. Using strategy rm

    3. Applied flip-+32.7

      \[\leadsto \]

    4. Simplified16.7

      \[\leadsto \]

    5. Simplified16.7

      \[\leadsto \]
    6. Using strategy rm

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

      \[\leadsto \]

    8. Applied times-frac16.7

      \[\leadsto \]

    9. Applied times-frac16.6

      \[\leadsto \]

    10. Simplified16.6

      \[\leadsto \]

    11. Simplified9.1

      \[\leadsto \]

    if 2.35376843954459236e133 < b

    1. Initial program 62.0

      \[\]

    2. Taylor expanded around inf 2.1

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

  4. Final simplification6.9

    \[\leadsto \]

Reproduce

herbie shell --seed 2020190 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))