Average Error: 27.8 → 28.2
Time: 36.5s
Precision: 64
Internal Precision: 128
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{1}{(\left((\left(a + y\right) \cdot y + b)_*\right) \cdot \left(y \cdot y\right) + \left((y \cdot c + i)_*\right))_*} \cdot (y \cdot \left((\left(y \cdot y\right) \cdot \left((y \cdot x + z)_*\right) + \left((y \cdot 27464.7644705 + 230661.510616)_*\right))_*\right) + t)_*\]

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 27.8

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

  2. Simplified28.1

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

  4. Applied div-inv28.2

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

  5. Final simplification28.2

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

Reproduce

herbie shell --seed 2019026 +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)))