Average Error: 5.7 → 4.3
Time: 25.2s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -6.488031990554541 \cdot 10^{-137}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 2.3865121378743047 \cdot 10^{-141}:\\ \;\;\;\;t \cdot \left(4.0 \cdot \left(-a\right)\right) + \left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \end{array}\]

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 < -6.488031990554541e-137 or 2.3865121378743047e-141 < t

    1. Initial program 3.3

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

    2. Simplified3.3

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.4

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

    if -6.488031990554541e-137 < t < 2.3865121378743047e-141

    1. Initial program 10.0

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

    2. Simplified9.8

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

    3. Taylor expanded around 0 6.1

      \[\leadsto \left(c \cdot b - \left(\left(27.0 \cdot j\right) \cdot k + \left(x \cdot 4.0\right) \cdot i\right)\right) + \left(\color{blue}{0} - a \cdot 4.0\right) \cdot t\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.488031990554541 \cdot 10^{-137}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 2.3865121378743047 \cdot 10^{-141}:\\ \;\;\;\;t \cdot \left(4.0 \cdot \left(-a\right)\right) + \left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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)))