The quadratic formula (r1)

Average Error: 33.6 → 7.1

Time: 5.4s

Precision: binary64

Math

\[\]

↓

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Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.5
Herbie	7.1

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Derivation

1. Split input into 4 regimes

2. if b < -9.13115209614994337e44

   1. Initial program 37.2

      \[\]

   2. Simplified 37.2

      \[\leadsto \]

   3. Taylor expanded around -inf 6.0

      \[\leadsto \]

   4. Simplified 6.0

      \[\leadsto \]

   if -9.13115209614994337e44 < b < -4.3571849990037956e-308

   1. Initial program 9.2

      \[\]

   2. Simplified 9.2

      \[\leadsto \]
   3. Using strategy rm

   4. Applied clear-num 9.3

      \[\leadsto \]

   5. Simplified 9.3

      \[\leadsto \]

   if -4.3571849990037956e-308 < b < 1.34309843668829568e38

   1. Initial program 28.3

      \[\]

   2. Simplified 28.3

      \[\leadsto \]
   3. Using strategy rm

   4. Applied flip-+ 28.4

      \[\leadsto \]

   5. Simplified 16.5

      \[\leadsto \]
   6. Using strategy rm

   7. Applied associate-/r* 16.5

      \[\leadsto \]

   8. Simplified 9.1

      \[\leadsto \]

   if 1.34309843668829568e38 < b

   1. Initial program 57.0

      \[\]

   2. Simplified 57.0

      \[\leadsto \]

   3. Taylor expanded around inf 4.2

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

4. Final simplification 7.1

   \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(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)))