Average Error: 5.5 → 3.7
Time: 1.4m
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.9153587237270055 \cdot 10^{+46}:\\ \;\;\;\;(\left(18.0 \cdot \left(\left(z \cdot x\right) \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{elif}\;y \le 1.383448850724911 \cdot 10^{-97}:\\ \;\;\;\;(\left(\left(18.0 \cdot x\right) \cdot \left(z \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(27.0 \cdot k\right) \cdot j + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(18.0 \cdot \left(\left(z \cdot x\right) \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \end{array}\]

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 y < -1.9153587237270055e+46 or 1.383448850724911e-97 < y

    1. Initial program 9.9

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

    2. Simplified11.2

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

    4. Applied *-un-lft-identity11.2

      \[\leadsto (\left(\left(y \cdot \color{blue}{\left(1 \cdot z\right)}\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t + \left(c \cdot b - (k \cdot \left(27.0 \cdot j\right) + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\]

    5. Applied associate-*r*11.2

      \[\leadsto (\left(\color{blue}{\left(\left(y \cdot 1\right) \cdot z\right)} \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t + \left(c \cdot b - (k \cdot \left(27.0 \cdot j\right) + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\]

    6. Applied associate-*l*6.3

      \[\leadsto (\left(\color{blue}{\left(y \cdot 1\right) \cdot \left(z \cdot \left(x \cdot 18.0\right)\right)} - a \cdot 4.0\right) \cdot t + \left(c \cdot b - (k \cdot \left(27.0 \cdot j\right) + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\]

    7. Simplified6.3

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

    8. Taylor expanded around inf 6.2

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

    9. Simplified6.2

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

    11. Applied associate-*r*6.2

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

    12. Applied associate-*r*6.2

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

    if -1.9153587237270055e+46 < y < 1.383448850724911e-97

    1. Initial program 1.3

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

    2. Simplified1.3

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

    3. Taylor expanded around inf 1.2

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

    4. Simplified1.3

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

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.9153587237270055 \cdot 10^{+46}:\\ \;\;\;\;(\left(18.0 \cdot \left(\left(z \cdot x\right) \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{elif}\;y \le 1.383448850724911 \cdot 10^{-97}:\\ \;\;\;\;(\left(\left(18.0 \cdot x\right) \cdot \left(z \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(27.0 \cdot k\right) \cdot j + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(18.0 \cdot \left(\left(z \cdot x\right) \cdot y\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +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)))