Average Error: 29.0 → 29.1
Time: 46.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\frac{(\left((y \cdot \left(\sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*} \cdot \left(\sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*} \cdot \sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*}\right)\right) + 230661.510616)_*\right) \cdot y + t)_*}{(y \cdot \left((\left((y \cdot \left(y + a\right) + b)_*\right) \cdot y + c)_*\right) + i)_*}\]

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 29.0

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

  2. Simplified29.0

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

  4. Applied *-un-lft-identity29.0

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

  5. Applied associate-/l*29.2

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

  7. Applied div-inv29.3

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

  8. Applied associate-/r*29.1

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

  10. Applied div-inv29.1

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

  11. Applied *-un-lft-identity29.1

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

  12. Applied times-frac29.1

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

  13. Simplified29.1

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

  14. Simplified29.0

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

  16. Applied add-cube-cbrt29.1

    \[\leadsto 1 \cdot \frac{(\left((y \cdot \color{blue}{\left(\left(\sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*} \cdot \sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*}\right) \cdot \sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*}\right)} + 230661.510616)_*\right) \cdot y + t)_*}{(y \cdot \left((\left((y \cdot \left(y + a\right) + b)_*\right) \cdot y + c)_*\right) + i)_*}\]

  17. Final simplification29.1

    \[\leadsto \frac{(\left((y \cdot \left(\sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*} \cdot \left(\sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*} \cdot \sqrt[3]{(y \cdot \left((y \cdot x + z)_*\right) + 27464.7644705)_*}\right)\right) + 230661.510616)_*\right) \cdot y + t)_*}{(y \cdot \left((\left((y \cdot \left(y + a\right) + b)_*\right) \cdot y + c)_*\right) + i)_*}\]

Reproduce

herbie shell --seed 2019053 +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)))