Average Error: 28.8 → 29.1
Time: 43.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{t + y \cdot (\left(y \cdot y\right) \cdot \left((y \cdot x + z)_*\right) + \left((y \cdot 27464.7644705 + 230661.510616)_*\right))_*}{(\left((\left(a + y\right) \cdot y + b)_*\right) \cdot \left(y \cdot y\right) + \left((y \cdot c + i)_*\right))_*}\]

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 28.8

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

  2. Simplified29.1

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

  4. Applied fma-udef29.1

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

  6. Applied *-commutative29.1

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

  7. Final simplification29.1

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

Reproduce

herbie shell --seed 2019030 +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)))