The quadratic formula (r1)

Average Error: 33.7 → 7.2

Time: 5.0s

Precision: binary64

Math

\[\]

↓

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Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.7
Target	20.9
Herbie	7.2

\[\]

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. Simplified 5.1

      \[\leadsto \]

   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. Simplified 16.7

      \[\leadsto \]

   5. Simplified 16.7

      \[\leadsto \]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.7

      \[\leadsto \]

   8. Applied times-frac 16.7

      \[\leadsto \]

   9. Applied times-frac 16.6

      \[\leadsto \]

   10. Simplified 16.6

       \[\leadsto \]

   11. Simplified 9.7

       \[\leadsto \]
   12. Using strategy rm

   13. Applied div-inv 9.8

       \[\leadsto \]

   14. Simplified 9.8

       \[\leadsto \]

   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 simplification 7.2

   \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

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