Average Error: 5.5 → 5.3
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.3990426501521596 \cdot 10^{-269}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right) + t \cdot \left(x \cdot \left(18.0 \cdot \left(y \cdot z\right)\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.7531631188330568 \cdot 10^{-225}:\\ \;\;\;\;\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - t \cdot \left(a \cdot 4.0\right)\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -1.3990426501521596e-269

    1. Initial program 4.8

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

    2. Initial simplification5.6

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

    4. Applied associate-*l*5.7

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

    if -1.3990426501521596e-269 < t < 1.7531631188330568e-225

    1. Initial program 10.0

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

    2. Initial simplification11.3

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

    3. Taylor expanded around 0 5.1

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

    if 1.7531631188330568e-225 < t

    1. Initial program 4.8

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.3990426501521596 \cdot 10^{-269}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right) + t \cdot \left(x \cdot \left(18.0 \cdot \left(y \cdot z\right)\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.7531631188330568 \cdot 10^{-225}:\\ \;\;\;\;\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - t \cdot \left(a \cdot 4.0\right)\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018221 
(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)))