NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 1.0

Time: 4.3s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 280.862230028215833

   1. Initial program 39.1

      \[\]

   2. Taylor expanded around 0 1.3

      \[\leadsto \]

   3. Simplified 1.3

      \[\leadsto \]
   4. Using strategy rm

   5. Applied add-cbrt-cube 1.3

      \[\leadsto \]

   6. Simplified 1.3

      \[\leadsto \]

   7. Taylor expanded around 0 1.3

      \[\leadsto \]

   8. Simplified 1.3

      \[\leadsto \]
   9. Using strategy rm

   10. Applied flip3-+ 1.3

       \[\leadsto \]

   11. Applied cbrt-div 1.3

       \[\leadsto \]

   12. Simplified 1.3

       \[\leadsto \]

   13. Simplified 1.3

       \[\leadsto \]

   if 280.862230028215833 < x

   1. Initial program 0.1

      \[\]
   2. Using strategy rm

   3. Applied exp-neg 0.1

      \[\leadsto \]

   4. Applied un-div-inv 0.1

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

4. Final simplification 1.0

   \[\leadsto \]

Reproduce

herbie shell --seed 2020190 
(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))