NMSE Section 6.1 mentioned, A

Average Error: 29.7 → 0.9

Time: 6.3s

Precision: binary64

Math

\[\]

↓

\[\]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 120.7427870229541

   1. Initial program 39.2

      \[\]

   2. Taylor expanded around 0 1.2

      \[\leadsto \]

   3. Simplified 1.2

      \[\leadsto \]

   4. Taylor expanded around 0 1.2

      \[\leadsto \]

   5. Simplified 1.2

      \[\leadsto \]
   6. Using strategy rm

   7. Applied add-cube-cbrt 1.2

      \[\leadsto \]

   8. Applied associate-*r* 1.2

      \[\leadsto \]
   9. Using strategy rm

   10. Applied cbrt-unprod 1.2

       \[\leadsto \]

   11. Simplified 1.2

       \[\leadsto \]

   if 120.7427870229541 < x

   1. Initial program 0.1

      \[\]
   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 0.9

   \[\leadsto \]

Reproduce

herbie shell --seed 2020191 
(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))