Average Error: 5.3 → 1.6
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot y \le -1.1649585003094704 \cdot 10^{+147} \lor \neg \left(z \cdot y \le 7.394863913131996 \cdot 10^{+29}\right):\\ \;\;\;\;(\left(\left(\left(x \cdot t\right) \cdot y\right) \cdot 18.0\right) \cdot z + \left((\left(-a\right) \cdot \left(4.0 \cdot t\right) + \left(c \cdot b\right))_*\right))_* - (i \cdot \left(4.0 \cdot x\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(z \cdot y\right) \cdot x\right) \cdot \left(18.0 \cdot t\right) + \left((\left(-a\right) \cdot \left(4.0 \cdot t\right) + \left(c \cdot b\right))_*\right))_* - (x \cdot \left(i \cdot 4.0\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\\ \end{array}\]

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 z) < -1.1649585003094704e+147 or 7.394863913131996e+29 < (* y z)

    1. Initial program 11.8

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

    2. Applied simplify3.6

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

    4. Applied associate-*r*3.4

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

    if -1.1649585003094704e+147 < (* y z) < 7.394863913131996e+29

    1. Initial program 2.7

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

    2. Taylor expanded around inf 2.7

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

    3. Applied simplify0.8

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

  4. Applied simplify1.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;z \cdot y \le -1.1649585003094704 \cdot 10^{+147} \lor \neg \left(z \cdot y \le 7.394863913131996 \cdot 10^{+29}\right):\\ \;\;\;\;(\left(\left(\left(x \cdot t\right) \cdot y\right) \cdot 18.0\right) \cdot z + \left((\left(-a\right) \cdot \left(4.0 \cdot t\right) + \left(c \cdot b\right))_*\right))_* - (i \cdot \left(4.0 \cdot x\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(z \cdot y\right) \cdot x\right) \cdot \left(18.0 \cdot t\right) + \left((\left(-a\right) \cdot \left(4.0 \cdot t\right) + \left(c \cdot b\right))_*\right))_* - (x \cdot \left(i \cdot 4.0\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018201 +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)))