Average Error: 5.1 → 3.1
Time: 18.1s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -11503630054022.404:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le -8.864908019360324 \cdot 10^{-251}:\\ \;\;\;\;\left(\left(\left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(z \cdot t\right) - t \cdot \left(a \cdot 4.0\right)\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.0456262008632395 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(\left(t \cdot a\right) \cdot \left(-4.0\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - 27.0 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 1.1572879973961506 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(z \cdot t\right) - t \cdot \left(a \cdot 4.0\right)\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \end{array}\]

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 3 regimes

  2. if t < -11503630054022.404 or 1.1572879973961506e+22 < t

    1. Initial program 1.7

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

    2. Simplified1.7

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

    if -11503630054022.404 < t < -8.864908019360324e-251 or 2.0456262008632395e-140 < t < 1.1572879973961506e+22

    1. Initial program 5.6

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

    3. Applied associate-*l*3.1

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

    if -8.864908019360324e-251 < t < 2.0456262008632395e-140

    1. Initial program 9.3

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

    3. Applied associate-*l*9.4

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

    4. Taylor expanded around inf 9.3

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

    5. Taylor expanded around inf 9.3

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

    6. Taylor expanded around 0 5.4

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -11503630054022.404:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le -8.864908019360324 \cdot 10^{-251}:\\ \;\;\;\;\left(\left(\left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(z \cdot t\right) - t \cdot \left(a \cdot 4.0\right)\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.0456262008632395 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(\left(t \cdot a\right) \cdot \left(-4.0\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - 27.0 \cdot \left(k \cdot j\right)\\ \mathbf{elif}\;t \le 1.1572879973961506 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(\left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(z \cdot t\right) - t \cdot \left(a \cdot 4.0\right)\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + k \cdot \left(j \cdot 27.0\right)\right)\right) + t \cdot \left(\left(\left(18.0 \cdot x\right) \cdot z\right) \cdot y - a \cdot 4.0\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 
(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)))