Toniolo and Linder, Equation (13)

Average Error: 34.6 → 32.4

Time: 30.8s

Precision: binary64

Math

\[\]

↓

\[\]

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 2 regimes

2. if n < -1.5437202692128072e-226 or -7.178830274298204e-295 < n

   1. Initial program 34.2

      \[\]

   2. Simplified 31.9

      \[\leadsto \]
   3. Using strategy rm

   4. Applied associate-*r* 31.6

      \[\leadsto \]
   5. Using strategy rm

   6. Applied associate-*r* 31.8

      \[\leadsto \]

   if -1.5437202692128072e-226 < n < -7.178830274298204e-295

   1. Initial program 39.5

      \[\]

   2. Simplified 38.2

      \[\leadsto \]
   3. Using strategy rm

   4. Applied associate-*r* 36.3

      \[\leadsto \]
   5. Using strategy rm

   6. Applied associate-*r* 37.4

      \[\leadsto \]

   7. Taylor expanded around inf 53.1

      \[\leadsto \]

   8. Simplified 38.7

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

4. Final simplification 32.4

   \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(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*))))))