Average Error: 5.4 → 5.0
Time: 48.9s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.0995503329171053 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 9.413052276743855 \cdot 10^{-271}:\\ \;\;\;\;\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c - \left(27.0 \cdot \left(j \cdot k\right) + \left(4.0 \cdot x\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - \left(\left(\sqrt[3]{27.0 \cdot \left(j \cdot k\right)} \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)}\right) \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)} + \left(4.0 \cdot x\right) \cdot i\right)\right) + \left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t\\ \end{array}\]

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 < -2.0995503329171053e-203

    1. Initial program 4.1

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

    if -2.0995503329171053e-203 < t < 9.413052276743855e-271

    1. Initial program 10.6

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

    2. Initial simplification11.4

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

    3. Taylor expanded around 0 5.7

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

    if 9.413052276743855e-271 < t

    1. Initial program 4.7

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

    2. Initial simplification5.1

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

    4. Applied add-cube-cbrt5.4

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.0995503329171053 \cdot 10^{-203}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 9.413052276743855 \cdot 10^{-271}:\\ \;\;\;\;\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c - \left(27.0 \cdot \left(j \cdot k\right) + \left(4.0 \cdot x\right) \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - \left(\left(\sqrt[3]{27.0 \cdot \left(j \cdot k\right)} \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)}\right) \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)} + \left(4.0 \cdot x\right) \cdot i\right)\right) + \left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right) - a \cdot 4.0\right) \cdot t\\ \end{array}\]

Runtime

Time bar (total: 48.9s)Debug logProfile

herbie shell --seed 2018258 
(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)))