Average Error: 5.0 → 4.3
Time: 56.2s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.3261342882246913 \cdot 10^{-137}:\\ \;\;\;\;(\left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{elif}\;t \le 1.2671208751780769 \cdot 10^{-132}:\\ \;\;\;\;(\left(4.0 \cdot \left(-a\right)\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(27.0 \cdot k\right) \cdot j + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \end{array}\]

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 < -3.3261342882246913e-137

    1. Initial program 3.1

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

    2. Simplified4.0

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

    3. Taylor expanded around -inf 3.9

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

    4. Simplified3.9

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

    6. Applied associate-*l*3.1

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

    if -3.3261342882246913e-137 < t < 1.2671208751780769e-132

    1. Initial program 8.5

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

    2. Simplified10.0

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

    3. Taylor expanded around -inf 10.0

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

    4. Simplified10.0

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

    5. Taylor expanded around 0 5.8

      \[\leadsto (\left(\color{blue}{0} - a \cdot 4.0\right) \cdot t + \left(c \cdot b - (\left(k \cdot j\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\]

    if 1.2671208751780769e-132 < t

    1. Initial program 3.0

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

    2. Simplified3.8

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

    5. Taylor expanded around -inf 3.7

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

    6. Simplified3.8

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

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.3261342882246913 \cdot 10^{-137}:\\ \;\;\;\;(\left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{elif}\;t \le 1.2671208751780769 \cdot 10^{-132}:\\ \;\;\;\;(\left(4.0 \cdot \left(-a\right)\right) \cdot t + \left(b \cdot c - (\left(j \cdot k\right) \cdot 27.0 + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t + \left(b \cdot c - (\left(27.0 \cdot k\right) \cdot j + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019090 +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)))