Average Error: 5.2 → 5.0
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.2139816614268297 \cdot 10^{-295} \lor \neg \left(t \le 4.455478615389272 \cdot 10^{-113}\right):\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(z \cdot y\right) \cdot x\right) \cdot 18.0\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(18.0 \cdot x\right) \cdot \left(z \cdot t\right)\right) + \left((\left(4.0 \cdot i\right) \cdot \left(-x\right) + \left(b \cdot c\right))_*\right))_* - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

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

Bits error versus j

Bits error versus k

Derivation

  1. Split input into 2 regimes

  2. if t < -1.2139816614268297e-295 or 4.455478615389272e-113 < t

    1. Initial program 4.3

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around -inf 5.0

      \[\leadsto \left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    if -1.2139816614268297e-295 < t < 4.455478615389272e-113

    1. Initial program 8.8

      \[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    2. Taylor expanded around -inf 9.6

      \[\leadsto \left(\left(\left(\color{blue}{\left(18.0 \cdot \left(x \cdot \left(z \cdot y\right)\right)\right)} \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    3. Taylor expanded around inf 13.5

      \[\leadsto \color{blue}{\left(\left(18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) + b \cdot c\right) - 4.0 \cdot \left(i \cdot x\right)\right)} - \left(j \cdot 27.0\right) \cdot k\]

    4. Simplified4.9

      \[\leadsto \color{blue}{(y \cdot \left(\left(18.0 \cdot x\right) \cdot \left(z \cdot t\right)\right) + \left((\left(i \cdot 4.0\right) \cdot \left(-x\right) + \left(b \cdot c\right))_*\right))_*} - \left(j \cdot 27.0\right) \cdot k\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.2139816614268297 \cdot 10^{-295} \lor \neg \left(t \le 4.455478615389272 \cdot 10^{-113}\right):\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(z \cdot y\right) \cdot x\right) \cdot 18.0\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(18.0 \cdot x\right) \cdot \left(z \cdot t\right)\right) + \left((\left(4.0 \cdot i\right) \cdot \left(-x\right) + \left(b \cdot c\right))_*\right))_* - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018251 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  (- (- (+ (- (* (* (* (* x 18.0) y) z) t) (* (* a 4.0) t)) (* b c)) (* (* x 4.0) i)) (* (* j 27.0) k)))