quadm (p42, negative)

Average Error: 34.4 → 6.6

Time: 5.6s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.4
Target	21.6
Herbie	6.6

\[\]

Derivation

1. Split input into 4 regimes

2. if b < -1.11280777925980924e121

   1. Initial program 61.2

      \[\]
   2. Using strategy rm

   3. Applied flip-- 61.2

      \[\leadsto \]

   4. Simplified 34.1

      \[\leadsto \]

   5. Simplified 34.1

      \[\leadsto \]
   6. Using strategy rm

   7. Applied *-un-lft-identity 34.1

      \[\leadsto \]

   8. Applied times-frac 34.1

      \[\leadsto \]

   9. Applied times-frac 34.1

      \[\leadsto \]

   10. Simplified 34.1

       \[\leadsto \]

   11. Simplified 32.8

       \[\leadsto \]

   12. Taylor expanded around -inf 6.2

       \[\leadsto \]

   13. Simplified 1.9

       \[\leadsto \]

   if -1.11280777925980924e121 < b < 7.34590175634682991e-281

   1. Initial program 32.3

      \[\]
   2. Using strategy rm

   3. Applied flip-- 32.3

      \[\leadsto \]

   4. Simplified 16.1

      \[\leadsto \]

   5. Simplified 16.1

      \[\leadsto \]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.1

      \[\leadsto \]

   8. Applied times-frac 16.1

      \[\leadsto \]

   9. Applied times-frac 16.1

      \[\leadsto \]

   10. Simplified 16.1

       \[\leadsto \]

   11. Simplified 8.8

       \[\leadsto \]

   if 7.34590175634682991e-281 < b < 4.9074896239589895e136

   1. Initial program 9.0

      \[\]
   2. Using strategy rm

   3. Applied div-inv 9.1

      \[\leadsto \]

   4. Simplified 9.1

      \[\leadsto \]

   if 4.9074896239589895e136 < b

   1. Initial program 58.2

      \[\]
   2. Using strategy rm

   3. Applied flip-- 63.8

      \[\leadsto \]

   4. Simplified 62.7

      \[\leadsto \]

   5. Simplified 62.7

      \[\leadsto \]

   6. Taylor expanded around 0 2.5

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

4. Final simplification 6.6

   \[\leadsto \]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))