Average Error: 5.1 → 4.2
Time: 33.9s
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.5911730309355884 \cdot 10^{-209}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + j \cdot \left(k \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.8151499724537247 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{elif}\;t \le 5.855155856274848 \cdot 10^{+247}:\\ \;\;\;\;t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) + \left(\sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)} \cdot \sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)}\right) \cdot \sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + j \cdot \left(k \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \end{array}\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -1.5911730309355884e-209 or 5.855155856274848e+247 < t

    1. Initial program 3.8

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

    2. Initial simplification4.9

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

    3. Taylor expanded around -inf 4.8

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

    4. Simplified4.0

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

    6. Applied associate-*r*4.1

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

    if -1.5911730309355884e-209 < t < 1.8151499724537247e+59

    1. Initial program 7.1

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

    3. Applied associate-*l*4.7

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

    if 1.8151499724537247e+59 < t < 5.855155856274848e+247

    1. Initial program 1.3

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

    2. Initial simplification1.6

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

    4. Applied add-cube-cbrt2.1

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.5911730309355884 \cdot 10^{-209}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + j \cdot \left(k \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \mathbf{elif}\;t \le 1.8151499724537247 \cdot 10^{+59}:\\ \;\;\;\;\left(\left(c \cdot b + \left(\left(\left(18.0 \cdot x\right) \cdot y\right) \cdot \left(t \cdot z\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - i \cdot \left(x \cdot 4.0\right)\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{elif}\;t \le 5.855155856274848 \cdot 10^{+247}:\\ \;\;\;\;t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right) - a \cdot 4.0\right) + \left(\sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)} \cdot \sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)}\right) \cdot \sqrt[3]{c \cdot b - \left(27.0 \cdot \left(j \cdot k\right) + i \cdot \left(x \cdot 4.0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot b - \left(i \cdot \left(x \cdot 4.0\right) + j \cdot \left(k \cdot 27.0\right)\right)\right) + t \cdot \left(\left(x \cdot z\right) \cdot \left(y \cdot 18.0\right) - a \cdot 4.0\right)\\ \end{array}\]

Runtime

Time bar (total: 33.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes5.14.20.74.421%
herbie shell --seed 2018339 
(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)))