Average Error: 33.0 → 6.8
Time: 6.6s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -1.0012485150414439e43

    1. Initial program 35.8

      \[\]

    2. Taylor expanded around -inf 6.0

      \[\leadsto \]

    if -1.0012485150414439e43 < b_2 < -2.37539736433464885e-198

    1. Initial program 6.2

      \[\]
    2. Using strategy rm

    3. Applied div-inv6.4

      \[\leadsto \]

    if -2.37539736433464885e-198 < b_2 < 5.8169366956900916e72

    1. Initial program 27.6

      \[\]
    2. Using strategy rm

    3. Applied flip-+27.8

      \[\leadsto \]

    4. Simplified16.1

      \[\leadsto \]
    5. Using strategy rm

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

      \[\leadsto \]

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

      \[\leadsto \]

    8. Applied times-frac16.1

      \[\leadsto \]

    9. Simplified16.1

      \[\leadsto \]

    10. Simplified10.2

      \[\leadsto \]

    if 5.8169366956900916e72 < b_2

    1. Initial program 57.4

      \[\]
    2. Using strategy rm

    3. Applied flip-+57.4

      \[\leadsto \]

    4. Simplified29.2

      \[\leadsto \]
    5. Using strategy rm

    6. Applied *-un-lft-identity29.2

      \[\leadsto \]

    7. Applied *-un-lft-identity29.2

      \[\leadsto \]

    8. Applied times-frac29.2

      \[\leadsto \]

    9. Simplified29.2

      \[\leadsto \]

    10. Simplified26.2

      \[\leadsto \]

    11. Taylor expanded around inf 7.2

      \[\leadsto \]

    12. Simplified3.4

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

  4. Final simplification6.8

    \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (neg b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))