Average Error: 34.8 → 28.3
Time: 22.9s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

  1. Split input into 3 regimes

  2. if n < -1.09727602493234047e-71

    1. Initial program 31.4

      \[\]

    2. Simplified29.8

      \[\leadsto \]
    3. Using strategy rm

    4. Applied sqr-pow29.8

      \[\leadsto \]

    5. Applied associate-*l*28.5

      \[\leadsto \]

    6. Simplified28.5

      \[\leadsto \]
    7. Using strategy rm

    8. Applied associate-*r*27.6

      \[\leadsto \]

    if -1.09727602493234047e-71 < n < 4.27893538693138e-311

    1. Initial program 38.1

      \[\]

    2. Simplified35.8

      \[\leadsto \]
    3. Using strategy rm

    4. Applied sqr-pow35.8

      \[\leadsto \]

    5. Applied associate-*l*35.3

      \[\leadsto \]

    6. Simplified35.3

      \[\leadsto \]

    7. Taylor expanded around inf 52.1

      \[\leadsto \]

    8. Simplified34.9

      \[\leadsto \]

    if 4.27893538693138e-311 < n

    1. Initial program 34.8

      \[\]

    2. Simplified32.9

      \[\leadsto \]
    3. Using strategy rm

    4. Applied sqrt-prod33.1

      \[\leadsto \]
    5. Using strategy rm

    6. Applied sqrt-prod25.5

      \[\leadsto \]

    7. Simplified25.5

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

  4. Final simplification28.3

    \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))