Average Error: 5.2 → 3.3
Time: 30.9s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.904235549875742 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot \left(x \cdot y\right)\right) \cdot 18.0\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 6.13092242918685 \cdot 10^{-07}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot t\right) \cdot \left(y \cdot \left(18.0 \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

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 < -4.904235549875742e+79

    1. Initial program 1.6

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

    2. Taylor expanded around inf 1.6

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

    4. Applied associate-*l*1.6

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

    if -4.904235549875742e+79 < t < 6.13092242918685e-07

    1. Initial program 6.8

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

    2. Taylor expanded around inf 6.8

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

    4. Applied associate-*l*4.0

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

    if 6.13092242918685e-07 < t

    1. Initial program 1.5

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

    2. Taylor expanded around -inf 1.8

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.904235549875742 \cdot 10^{+79}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot \left(x \cdot y\right)\right) \cdot 18.0\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 6.13092242918685 \cdot 10^{-07}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(z \cdot t\right) \cdot \left(y \cdot \left(18.0 \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(t \cdot \left(\left(z \cdot y\right) \cdot x\right)\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \end{array}\]

Runtime

Time bar (total: 30.9s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes5.13.30.74.541.5%
herbie shell --seed 2018354 +o rules:numerics
(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)))