Average Error: 28.2 → 10.4
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.0100253264217397 \cdot 10^{+67} \lor \neg \left(y \le 3.756033471922753 \cdot 10^{+52}\right):\\ \;\;\;\;x + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{(y \cdot \left((\left(y \cdot y\right) \cdot \left((y \cdot x + z)_*\right) + \left((y \cdot 27464.7644705 + 230661.510616)_*\right))_*\right) + t)_*}{(\left(y \cdot y\right) \cdot \left((\left(a + y\right) \cdot y + b)_*\right) + \left((y \cdot c + i)_*\right))_*}\\ \end{array}\]

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. Split input into 2 regimes

  2. if y < -1.0100253264217397e+67 or 3.756033471922753e+52 < y

    1. Initial program 61.3

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

    2. Initial simplification61.3

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

    4. Applied add-cube-cbrt61.3

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

    5. Taylor expanded around inf 17.6

      \[\leadsto \color{blue}{x + \frac{z}{y}}\]

    if -1.0100253264217397e+67 < y < 3.756033471922753e+52

    1. Initial program 4.7

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

    2. Initial simplification5.3

      \[\leadsto \frac{(y \cdot \left((\left(y \cdot y\right) \cdot \left((y \cdot x + z)_*\right) + \left((y \cdot 27464.7644705 + 230661.510616)_*\right))_*\right) + t)_*}{(\left(y \cdot y\right) \cdot \left((\left(y + a\right) \cdot y + b)_*\right) + \left((y \cdot c + i)_*\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0100253264217397 \cdot 10^{+67} \lor \neg \left(y \le 3.756033471922753 \cdot 10^{+52}\right):\\ \;\;\;\;x + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{(y \cdot \left((\left(y \cdot y\right) \cdot \left((y \cdot x + z)_*\right) + \left((y \cdot 27464.7644705 + 230661.510616)_*\right))_*\right) + t)_*}{(\left(y \cdot y\right) \cdot \left((\left(a + y\right) \cdot y + b)_*\right) + \left((y \cdot c + i)_*\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018216 +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)))