VandenBroeck and Keller, Equation (23)

Average Error: 13.9 → 1.5

Time: 15.4s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus F

Bits error versus B

Bits error versus x

Derivation

1. Split input into 3 regimes

2. if F < -1.3553598376870535e154

   1. Initial program 41.5

      \[\]

   2. Simplified 36.4

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 36.4

      \[\leadsto \]

   5. Applied associate-*r* 36.4

      \[\leadsto \]

   6. Simplified 36.4

      \[\leadsto \]
   7. Using strategy rm

   8. Applied distribute-frac-neg 36.4

      \[\leadsto \]

   9. Applied pow-neg 36.4

      \[\leadsto \]

   10. Applied un-div-inv 36.4

       \[\leadsto \]

   11. Taylor expanded around -inf 3.9

       \[\leadsto \]

   12. Simplified 4.2

       \[\leadsto \]

   if -1.3553598376870535e154 < F < 1.96833105963148191e142

   1. Initial program 2.2

      \[\]

   2. Simplified 0.4

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 0.4

      \[\leadsto \]

   5. Applied associate-*r* 0.4

      \[\leadsto \]

   6. Simplified 0.4

      \[\leadsto \]
   7. Using strategy rm

   8. Applied distribute-frac-neg 0.4

      \[\leadsto \]

   9. Applied pow-neg 0.4

      \[\leadsto \]

   10. Applied un-div-inv 0.3

       \[\leadsto \]

   if 1.96833105963148191e142 < F

   1. Initial program 40.0

      \[\]

   2. Simplified 34.5

      \[\leadsto \]
   3. Using strategy rm

   4. Applied div-inv 34.5

      \[\leadsto \]

   5. Applied associate-*r* 34.5

      \[\leadsto \]

   6. Simplified 34.5

      \[\leadsto \]
   7. Using strategy rm

   8. Applied distribute-frac-neg 34.5

      \[\leadsto \]

   9. Applied pow-neg 34.5

      \[\leadsto \]

   10. Applied un-div-inv 34.5

       \[\leadsto \]

   11. Taylor expanded around inf 3.9

       \[\leadsto \]

   12. Simplified 4.2

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

4. Final simplification 1.5

   \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (neg (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (neg (/ 1.0 2.0))))))