Average Error: 28.3 → 28.4
Time: 1.6m
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*} \cdot (y \cdot \left((y \cdot \left((\left((x \cdot y + z)_*\right) \cdot y + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 28.3

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified28.3

    \[\leadsto \color{blue}{\frac{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*}}\]
  3. Using strategy rm

  4. Applied clear-num28.5

    \[\leadsto \color{blue}{\frac{1}{\frac{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}}\]
  5. Using strategy rm

  6. Applied div-inv28.6

    \[\leadsto \frac{1}{\color{blue}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_* \cdot \frac{1}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity28.6

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_* \cdot \frac{1}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}\]

  9. Applied times-frac28.4

    \[\leadsto \color{blue}{\frac{1}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*} \cdot \frac{1}{\frac{1}{(y \cdot \left((y \cdot \left((y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}}}\]

  10. Simplified28.4

    \[\leadsto \frac{1}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*} \cdot \color{blue}{(y \cdot \left((y \cdot \left((\left((x \cdot y + z)_*\right) \cdot y + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*}\]

  11. Final simplification28.4

    \[\leadsto \frac{1}{(\left((y \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + c)_*\right) \cdot y + i)_*} \cdot (y \cdot \left((y \cdot \left((\left((x \cdot y + z)_*\right) \cdot y + 27464.7644705)_*\right) + 230661.510616)_*\right) + t)_*\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))