VandenBroeck and Keller, Equation (23)

Average Error: 13.4 → 1.3

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 < -1.33081947572925e154

   1. Initial program 41.6

      \[\]

   2. Simplified 36.8

      \[\leadsto \]
   3. Using strategy rm

   4. Applied associate-*r/ 36.8

      \[\leadsto \]

   5. Simplified 36.8

      \[\leadsto \]

   6. Taylor expanded around -inf 4.0

      \[\leadsto \]

   7. Simplified 4.0

      \[\leadsto \]

   if -1.33081947572925e154 < F < 1.3658377657683982e154

   1. Initial program 2.6

      \[\]

   2. Simplified 0.4

      \[\leadsto \]
   3. Using strategy rm

   4. Applied associate-*r/ 0.3

      \[\leadsto \]

   5. Simplified 0.3

      \[\leadsto \]
   6. Using strategy rm

   7. Applied distribute-frac-neg 0.3

      \[\leadsto \]

   8. Applied pow-neg 0.4

      \[\leadsto \]

   9. Applied un-div-inv 0.3

      \[\leadsto \]
   10. Using strategy rm

   11. Applied associate-*r/ 0.2

       \[\leadsto \]

   if 1.3658377657683982e154 < F

   1. Initial program 39.8

      \[\]

   2. Simplified 34.7

      \[\leadsto \]
   3. Using strategy rm

   4. Applied associate-*r/ 34.7

      \[\leadsto \]

   5. Simplified 34.7

      \[\leadsto \]

   6. Taylor expanded around inf 3.8

      \[\leadsto \]

   7. Simplified 3.8

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

4. Final simplification 1.3

   \[\leadsto \]

Reproduce

herbie shell --seed 2020180 
(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))))))