NMSE Section 6.1 mentioned, A

Average Error: 29.5 → 0.9

Time: 5.2s

Precision: binary64

• could not determine a ground truth for program body

  1. x = -2.9943033005252086e+246
  2. eps = -1.3448709461475583e-297

Math

\[\]

↓

\[\]

Error

Bits error versus x

Bits error versus eps

Derivation

1. Split input into 2 regimes

2. if x < 296.43862586906903

   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

      \[\leadsto \]

   if 296.43862586906903 < x

   1. Initial program 0.0

      \[\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.0

      \[\leadsto \]

   4. Simplified 0.0

      \[\leadsto \]

   5. Simplified 0.0

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

4. Final simplification 0.9

   \[\leadsto \]

Reproduce

herbie shell --seed 2020192 
(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))