sqrt A

Average Error: 30.7 → 0.3

Time: 1.7s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -8.44615837323415e-311

   1. Initial program 30.8

      \[\]

   2. Simplified 30.8

      \[\leadsto \]

   3. Taylor expanded around -inf 0.4

      \[\leadsto \]

   4. Simplified 0.4

      \[\leadsto \]

   if -8.44615837323415e-311 < x

   1. Initial program 30.7

      \[\]

   2. Simplified 30.7

      \[\leadsto \]

   3. Taylor expanded around 0 0.4

      \[\leadsto \]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 0.4

      \[\leadsto \]

   6. Applied sqrt-prod 0.6

      \[\leadsto \]

   7. Applied associate-*r* 0.4

      \[\leadsto \]
   8. Using strategy rm

   9. Applied add-sqr-sqrt 0.4

      \[\leadsto \]

   10. Applied sqrt-prod 0.4

       \[\leadsto \]

   11. Applied sqrt-prod 0.4

       \[\leadsto \]

   12. Applied associate-*r* 0.4

       \[\leadsto \]

   13. Simplified 0.3

       \[\leadsto \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))