Average Error: 30.9 → 0.3
Time: 1.6s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 8.31046261602783e-311

    1. Initial program 30.7

      \[\]

    2. Simplified30.7

      \[\leadsto \]

    3. Taylor expanded around -inf 0.4

      \[\leadsto \]

    4. Simplified0.4

      \[\leadsto \]

    if 8.31046261602783e-311 < x

    1. Initial program 31.1

      \[\]

    2. Simplified31.1

      \[\leadsto \]

    3. Taylor expanded around 0 0.4

      \[\leadsto \]
    4. Using strategy rm

    5. Applied add-sqr-sqrt0.4

      \[\leadsto \]

    6. Applied sqrt-prod0.6

      \[\leadsto \]

    7. Applied associate-*r*0.4

      \[\leadsto \]
    8. Using strategy rm

    9. Applied add-sqr-sqrt0.4

      \[\leadsto \]

    10. Applied sqrt-prod0.4

      \[\leadsto \]

    11. Applied sqrt-prod0.4

      \[\leadsto \]

    12. Applied associate-*r*0.4

      \[\leadsto \]

    13. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(FPCore (x)
  :name "sqrt A"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))