Average Error: 5.2 → 3.4
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.7951353429941762 \cdot 10^{+50}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.5773705127983518 \cdot 10^{-112}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(18.0 \cdot \left(\left(y \cdot z\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

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 < -2.7951353429941762e+50

    1. Initial program 1.7

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

    2. Initial simplification2.1

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

    3. Taylor expanded around inf 2.0

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

    4. Simplified1.2

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

    if -2.7951353429941762e+50 < t < 1.5773705127983518e-112

    1. Initial program 7.4

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

    3. Applied associate-*l*4.2

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

    if 1.5773705127983518e-112 < t

    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. Taylor expanded around -inf 2.8

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

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.7951353429941762 \cdot 10^{+50}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + \left(k \cdot j\right) \cdot 27.0\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.5773705127983518 \cdot 10^{-112}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot \left(18.0 \cdot \left(\left(y \cdot z\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018251 
(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)))