Average Error: 5.2 → 4.7
Time: 22.0s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.3281706519192746 \cdot 10^{-149}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(\left(18.0 \cdot x\right) \cdot y\right)\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;x \le 3.3010610623093217 \cdot 10^{+18}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(18.0 \cdot x\right) \cdot y\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{elif}\;x \le 4.0984330114018746 \cdot 10^{+148}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0\right) \cdot t - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(18.0 \cdot x\right) \cdot y\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

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 x < 1.3281706519192746e-149

    1. Initial program 4.0

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

    3. Applied associate-*l*4.0

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

    4. Taylor expanded around inf 3.9

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

    if 1.3281706519192746e-149 < x < 3.3010610623093217e+18 or 4.0984330114018746e+148 < x

    1. Initial program 6.7

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

    3. Applied associate-*l*6.7

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

    5. Applied associate-*l*6.5

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

    if 3.3010610623093217e+18 < x < 4.0984330114018746e+148

    1. Initial program 9.5

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

    3. Applied associate-*l*9.5

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

    5. Applied associate-*l*9.7

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

    6. Taylor expanded around -inf 5.2

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

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.3281706519192746 \cdot 10^{-149}:\\ \;\;\;\;\left(\left(b \cdot c + \left(t \cdot \left(z \cdot \left(\left(18.0 \cdot x\right) \cdot y\right)\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;x \le 3.3010610623093217 \cdot 10^{+18}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(18.0 \cdot x\right) \cdot y\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{elif}\;x \le 4.0984330114018746 \cdot 10^{+148}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot \left(z \cdot y\right)\right) \cdot 18.0\right) \cdot t - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - j \cdot \left(27.0 \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(t \cdot z\right) \cdot \left(\left(18.0 \cdot x\right) \cdot y\right) - a \cdot \left(t \cdot 4.0\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

Reproduce

herbie shell --seed 2019068 +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)))