Average Error: 28.2 → 10.3
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.0100253264217397 \cdot 10^{+67} \lor \neg \left(y \le 3.756033471922753 \cdot 10^{+52}\right):\\ \;\;\;\;x + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right) \cdot y + 230661.510616\right) + t}{i + \left(\sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y} \cdot \sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y}\right) \cdot \sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y}}\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

    3. Applied add-cube-cbrt61.3

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

    4. Applied *-un-lft-identity61.3

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

    5. Applied times-frac61.3

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

    6. 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. Using strategy rm

    3. Applied add-cube-cbrt5.1

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.0100253264217397 \cdot 10^{+67} \lor \neg \left(y \le 3.756033471922753 \cdot 10^{+52}\right):\\ \;\;\;\;x + \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(\left(27464.7644705 + y \cdot \left(z + x \cdot y\right)\right) \cdot y + 230661.510616\right) + t}{i + \left(\sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y} \cdot \sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y}\right) \cdot \sqrt[3]{\left(y \cdot \left(b + \left(a + y\right) \cdot y\right) + c\right) \cdot y}}\\ \end{array}\]

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018216 
(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)))