Average Error: 5.4 → 2.3
Time: 30.6s
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.5699680065387263 \cdot 10^{-100}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;t \le 1.0758880150456309 \cdot 10^{-113}:\\ \;\;\;\;\left((\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x + \left(b \cdot c\right))_* - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(x \cdot 18.0\right) \cdot \left(z \cdot y\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(27.0 \cdot k\right) + \left(\left(4.0 \cdot x\right) \cdot i\right))_*\\ \end{array}\]

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

Derivation

  1. Split input into 3 regimes

  2. if t < -4.5699680065387263e-100

    1. Initial program 2.9

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

    3. Applied associate-*l*3.0

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

    4. Taylor expanded around 0 2.9

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

    if -4.5699680065387263e-100 < t < 1.0758880150456309e-113

    1. Initial program 8.8

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

    3. Applied associate-*l*8.9

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

    5. Applied associate-*l*4.6

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

    7. Applied *-un-lft-identity4.6

      \[\leadsto \left(\color{blue}{1 \cdot \left(\left(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right)} - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]

    8. Applied prod-diff4.6

      \[\leadsto \color{blue}{\left((1 \cdot \left(\left(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(z \cdot t\right) - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) + \left(-i \cdot \left(x \cdot 4.0\right)\right))_* + (\left(-i\right) \cdot \left(x \cdot 4.0\right) + \left(i \cdot \left(x \cdot 4.0\right)\right))_*\right)} - \left(j \cdot 27.0\right) \cdot k\]

    9. Simplified1.1

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

    10. Simplified1.1

      \[\leadsto \left(\left((\left(\left(18.0 \cdot y\right) \cdot \left(t \cdot z\right)\right) \cdot x + \left(b \cdot c\right))_* - 4.0 \cdot \left(t \cdot a + i \cdot x\right)\right) + \color{blue}{0}\right) - \left(j \cdot 27.0\right) \cdot k\]

    if 1.0758880150456309e-113 < t

    1. Initial program 3.0

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

    2. Initial simplification3.4

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

  4. Final simplification2.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.5699680065387263 \cdot 10^{-100}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \mathbf{elif}\;t \le 1.0758880150456309 \cdot 10^{-113}:\\ \;\;\;\;\left((\left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) \cdot x + \left(b \cdot c\right))_* - \left(a \cdot t + i \cdot x\right) \cdot 4.0\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(x \cdot 18.0\right) \cdot \left(z \cdot y\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(27.0 \cdot k\right) + \left(\left(4.0 \cdot x\right) \cdot i\right))_*\\ \end{array}\]

Runtime

Time bar (total: 30.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes5.22.30.15.157.2%
herbie shell --seed 2018290 +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)))