Average Error: 5.2 → 3.0
Time: 54.7s
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot z \le -8.753706660248199 \cdot 10^{+36}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{if}\;y \cdot z \le 9.488642198219068 \cdot 10^{+166}:\\ \;\;\;\;(\left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right)\right) \cdot t + \left(b \cdot c\right))_* - (4.0 \cdot \left((t \cdot a + \left(x \cdot i\right))_*\right) + \left(\left(27.0 \cdot k\right) \cdot j\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(z \cdot 18.0\right) \cdot \left(t \cdot \left(y \cdot x\right)\right) + \left(c \cdot b\right))_* - (4.0 \cdot \left((a \cdot t + \left(x \cdot i\right))_*\right) + \left(\left(k \cdot j\right) \cdot 27.0\right))_*\\ \end{array}\]

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 (* y z) < -8.753706660248199e+36

    1. Initial program 10.7

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

    if -8.753706660248199e+36 < (* y z) < 9.488642198219068e+166

    1. Initial program 2.8

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

    2. Applied simplify0.8

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

    if 9.488642198219068e+166 < (* y z)

    1. Initial program 13.6

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

    2. Taylor expanded around inf 3.1

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

    3. Applied simplify5.1

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

Runtime

Time bar (total: 54.7s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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)))