Average Error: 30.3 → 0.3
Time: 1.8s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 3.59275856969827e-310

    1. Initial program 30.0

      \[\]

    2. Simplified30.0

      \[\leadsto \]

    3. Taylor expanded around -inf 0.4

      \[\leadsto \]

    4. Simplified0.4

      \[\leadsto \]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.4

      \[\leadsto \]

    7. Applied distribute-lft-neg-in0.4

      \[\leadsto \]

    8. Applied associate-*r*0.4

      \[\leadsto \]

    9. Simplified0.4

      \[\leadsto \]
    10. Using strategy rm

    11. Applied add-sqr-sqrt0.4

      \[\leadsto \]

    12. Applied associate-*r*0.7

      \[\leadsto \]

    13. Simplified0.3

      \[\leadsto \]

    if 3.59275856969827e-310 < x

    1. Initial program 30.6

      \[\]

    2. Simplified30.6

      \[\leadsto \]
    3. Using strategy rm

    4. Applied sqrt-prod0.3

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

  4. Final simplification0.3

    \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))