Average Error: 5.3 → 4.1
Time: 17.7s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -7.943170851973496 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(18.0 \cdot \left(x \cdot z\right)\right) \cdot y\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 5.952569630518274 \cdot 10^{-233}:\\ \;\;\;\;\left(\left(b \cdot c + \left(-t\right) \cdot \left(a \cdot 4.0\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot \left(\left(x \cdot z\right) \cdot y\right)\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - \left(k \cdot 27.0\right) \cdot j\\ \end{array}\]

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 < -7.943170851973496e-225

    1. Initial program 4.2

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

    2. Taylor expanded around inf 5.1

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

    4. Applied associate-*r*3.8

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

    5. Taylor expanded around inf 3.8

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

    7. Applied associate-*r*3.9

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

    if -7.943170851973496e-225 < t < 5.952569630518274e-233

    1. Initial program 9.9

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

    2. Taylor expanded around inf 10.9

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

    4. Applied associate-*r*9.0

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

    5. Taylor expanded around inf 9.0

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

    6. Taylor expanded around 0 4.2

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

    if 5.952569630518274e-233 < t

    1. Initial program 4.4

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

    2. Taylor expanded around inf 4.7

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

    4. Applied associate-*r*4.3

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

    5. Taylor expanded around inf 4.3

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

    7. Applied associate-*l*4.2

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

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -7.943170851973496 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(18.0 \cdot \left(x \cdot z\right)\right) \cdot y\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 5.952569630518274 \cdot 10^{-233}:\\ \;\;\;\;\left(\left(b \cdot c + \left(-t\right) \cdot \left(a \cdot 4.0\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot \left(\left(x \cdot z\right) \cdot y\right)\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - 4.0 \cdot \left(x \cdot i\right)\right) - \left(k \cdot 27.0\right) \cdot j\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(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)))