Average Error: 29.7 → 0.9
Time: 5.4s
Precision: binary64
\[\]
↓
\[\]

Error

Bits error versus x

Bits error versus eps

Derivation

  1. Split input into 2 regimes

  2. if x < 7.91027905227570916

    1. Initial program 39.4

      \[\]

    2. Taylor expanded around 0 1.0

      \[\leadsto \]

    3. Simplified1.0

      \[\leadsto \]
    4. Using strategy rm

    5. Applied add-cbrt-cube1.0

      \[\leadsto \]

    6. Applied add-cbrt-cube1.0

      \[\leadsto \]

    7. Applied add-cbrt-cube1.0

      \[\leadsto \]

    8. Applied cbrt-unprod1.0

      \[\leadsto \]

    9. Applied cbrt-unprod1.0

      \[\leadsto \]

    10. Simplified1.0

      \[\leadsto \]
    11. Using strategy rm

    12. Applied add-cube-cbrt1.0

      \[\leadsto \]
    13. Using strategy rm

    14. Applied flip3--1.0

      \[\leadsto \]

    15. Applied cube-div1.0

      \[\leadsto \]

    16. Applied associate-*r/1.0

      \[\leadsto \]

    17. Applied cbrt-div1.0

      \[\leadsto \]

    18. Applied associate-*r/1.0

      \[\leadsto \]

    19. Applied cbrt-div1.0

      \[\leadsto \]

    20. Simplified1.0

      \[\leadsto \]

    21. Simplified1.0

      \[\leadsto \]

    if 7.91027905227570916 < x

    1. Initial program 0.5

      \[\]
    2. Using strategy rm

    3. Applied add-cube-cbrt0.5

      \[\leadsto \]

    4. Simplified0.5

      \[\leadsto \]

    5. Simplified0.5

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

  4. Final simplification0.9

    \[\leadsto \]

Reproduce

herbie shell --seed 2020181 
(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))