quad2p (problem 3.2.1, positive)

Average Error: 34.5 → 6.9

Time: 5.3s

Precision: binary64

Math

\[\]

↓

\[\]

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 < -1.8605498001882586e27

   1. Initial program 34.5

      \[\]

   2. Simplified 34.5

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 34.6

      \[\leadsto \]

   5. Taylor expanded around -inf 6.2

      \[\leadsto \]

   6. Simplified 6.2

      \[\leadsto \]

   if -1.8605498001882586e27 < b_2 < -3.670160880532066e-242

   1. Initial program 8.8

      \[\]

   2. Simplified 8.8

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 9.0

      \[\leadsto \]

   if -3.670160880532066e-242 < b_2 < 3.6240060206103599e133

   1. Initial program 32.2

      \[\]

   2. Simplified 32.2

      \[\leadsto \]
   3. Using strategy rm

   4. Applied flip-- 32.3

      \[\leadsto \]

   5. Simplified 16.3

      \[\leadsto \]

   6. Simplified 16.3

      \[\leadsto \]
   7. Using strategy rm

   8. Applied distribute-rgt-neg-out 16.3

      \[\leadsto \]

   9. Applied distribute-frac-neg 16.3

      \[\leadsto \]

   10. Applied distribute-frac-neg 16.3

       \[\leadsto \]

   11. Simplified 9.0

       \[\leadsto \]

   if 3.6240060206103599e133 < b_2

   1. Initial program 62.1

      \[\]

   2. Simplified 62.1

      \[\leadsto \]
   3. Using strategy rm

   4. Applied flip-- 62.1

      \[\leadsto \]

   5. Simplified 35.4

      \[\leadsto \]

   6. Simplified 35.4

      \[\leadsto \]
   7. Using strategy rm

   8. Applied distribute-rgt-neg-out 35.4

      \[\leadsto \]

   9. Applied distribute-frac-neg 35.4

      \[\leadsto \]

   10. Applied distribute-frac-neg 35.4

       \[\leadsto \]

   11. Simplified 34.2

       \[\leadsto \]

   12. Taylor expanded around inf 6.5

       \[\leadsto \]

   13. Simplified 2.0

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

4. Final simplification 6.9

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(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))