NMSE Section 6.1 mentioned, A

Average Error: 29.9 → 1.0

Time: 4.7s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 246.92990390970124

   1. Initial program 39.7

      \[\]

   2. Taylor expanded around 0 1.2

      \[\leadsto \]

   3. Simplified 1.2

      \[\leadsto \]
   4. Using strategy rm

   5. Applied flip-- 1.2

      \[\leadsto \]

   6. Applied associate-*r/ 1.2

      \[\leadsto \]

   7. Simplified 1.2

      \[\leadsto \]
   8. Using strategy rm

   9. Applied add-cube-cbrt 1.2

      \[\leadsto \]

   10. Applied associate-*l* 1.2

       \[\leadsto \]

   11. Simplified 1.2

       \[\leadsto \]
   12. Using strategy rm

   13. Applied flip3-- 1.2

       \[\leadsto \]

   14. Applied associate-*r/ 1.2

       \[\leadsto \]

   15. Applied associate-*r/ 1.2

       \[\leadsto \]

   16. Applied associate-/l/ 1.2

       \[\leadsto \]

   17. Simplified 1.2

       \[\leadsto \]

   if 246.92990390970124 < x

   1. Initial program 0.2

      \[\]
   2. Using strategy rm

   3. Applied exp-neg 0.2

      \[\leadsto \]

   4. Applied un-div-inv 0.2

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

4. Final simplification 1.0

   \[\leadsto \]

Reproduce

herbie shell --seed 2020179 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (neg (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (neg (* (+ 1.0 eps) x))))) 2.0))