Average Error: 34.0 → 6.8
Time: 5.2s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.4
Herbie6.8
\[\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.8895171073387026e157

    1. Initial program 64.0

      \[\]

    2. Taylor expanded around -inf 1.5

      \[\leadsto \]

    3. Simplified1.5

      \[\leadsto \]

    if -3.8895171073387026e157 < b < 1.125215516807859e-284

    1. Initial program 34.4

      \[\]
    2. Using strategy rm

    3. Applied flip--34.4

      \[\leadsto \]

    4. Simplified16.3

      \[\leadsto \]

    5. Simplified16.3

      \[\leadsto \]
    6. Using strategy rm

    7. Applied *-un-lft-identity16.3

      \[\leadsto \]

    8. Applied times-frac16.3

      \[\leadsto \]

    9. Applied times-frac16.2

      \[\leadsto \]

    10. Simplified16.2

      \[\leadsto \]

    11. Simplified8.0

      \[\leadsto \]

    if 1.125215516807859e-284 < b < 1.6926019214523237e95

    1. Initial program 9.5

      \[\]
    2. Using strategy rm

    3. Applied div-inv9.6

      \[\leadsto \]

    4. Simplified9.6

      \[\leadsto \]

    if 1.6926019214523237e95 < b

    1. Initial program 45.3

      \[\]

    2. Taylor expanded around inf 10.8

      \[\leadsto \]

    3. Simplified4.7

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

  4. Final simplification6.8

    \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (neg b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))