Average Error: 5.5 → 4.0
Time: 22.1s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.667025378133879 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot z\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le -2.8179660425504494 \cdot 10^{-204}:\\ \;\;\;\;\left((\left(\left(z \cdot y\right) \cdot \left(x \cdot t\right)\right) \cdot 18.0 + \left(b \cdot c\right))_* - 4.0 \cdot \left(a \cdot t + x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.2124941510274635 \cdot 10^{-135}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(\left(\left(x \cdot z\right) \cdot t\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(z \cdot y\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(\left(4.0 \cdot x\right) \cdot i\right))_*\\ \end{array}\]

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 4 regimes

  2. if t < -1.667025378133879e-20

    1. Initial program 1.8

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

    2. Taylor expanded around inf 2.0

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

    4. Applied associate-*r*1.5

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

    6. Applied associate-*r*1.6

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

    if -1.667025378133879e-20 < t < -2.8179660425504494e-204

    1. Initial program 6.3

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

    2. Taylor expanded around inf 7.6

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

    4. Applied associate-*r*6.4

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

    6. Applied associate-*r*6.4

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

    8. Applied *-un-lft-identity6.4

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

    9. Applied prod-diff6.4

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

    10. Simplified5.1

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

    11. Simplified5.1

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

    if -2.8179660425504494e-204 < t < 2.2124941510274635e-135

    1. Initial program 9.7

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

    2. Taylor expanded around inf 10.2

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

    4. Applied associate-*r*9.2

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

    6. Applied associate-*r*4.9

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

    if 2.2124941510274635e-135 < t

    1. Initial program 3.3

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

    2. Simplified3.9

      \[\leadsto \color{blue}{(t \cdot \left(\left(y \cdot z\right) \cdot \left(x \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(\left(x \cdot 4.0\right) \cdot i\right))_*}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.667025378133879 \cdot 10^{-20}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(18.0 \cdot t\right) \cdot \left(\left(x \cdot z\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le -2.8179660425504494 \cdot 10^{-204}:\\ \;\;\;\;\left((\left(\left(z \cdot y\right) \cdot \left(x \cdot t\right)\right) \cdot 18.0 + \left(b \cdot c\right))_* - 4.0 \cdot \left(a \cdot t + x \cdot i\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.2124941510274635 \cdot 10^{-135}:\\ \;\;\;\;\left(\left(b \cdot c + \left(18.0 \cdot \left(\left(\left(x \cdot z\right) \cdot t\right) \cdot y\right) - \left(a \cdot 4.0\right) \cdot t\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(z \cdot y\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(\left(4.0 \cdot x\right) \cdot i\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019030 +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)))