Average Error: 33.7 → 7.2
Time: 5.0s
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 < -1.05732335807039929e75

    1. Initial program 41.6

      \[\]

    2. Taylor expanded around -inf 5.1

      \[\leadsto \]

    3. Simplified5.1

      \[\leadsto \]

    if -1.05732335807039929e75 < b < -5.43716846905727293e-278

    1. Initial program 9.4

      \[\]

    if -5.43716846905727293e-278 < b < 2.09269847736041385e71

    1. Initial program 30.4

      \[\]
    2. Using strategy rm

    3. Applied flip-+30.4

      \[\leadsto \]

    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. Simplified9.7

      \[\leadsto \]
    12. Using strategy rm

    13. Applied div-inv9.8

      \[\leadsto \]

    14. Simplified9.8

      \[\leadsto \]

    if 2.09269847736041385e71 < b

    1. Initial program 57.7

      \[\]

    2. Taylor expanded around inf 3.3

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

  4. Final simplification7.2

    \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))