Average Error: 5.5 → 4.8
Time: 1.6m
Precision: 64
Internal Precision: 320
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;j \le -1.638603102240291 \cdot 10^{-07}:\\ \;\;\;\;(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(t \cdot z\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* + \left(\left(j \cdot k\right) \cdot \left(-27.0\right) - i \cdot \left(4.0 \cdot x\right)\right)\\ \mathbf{elif}\;j \le 7.036950341320534 \cdot 10^{-163}:\\ \;\;\;\;(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(18.0 \cdot \left(x \cdot t\right)\right) \cdot z\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(x \cdot \left(4.0 \cdot i\right)\right))_*\\ \end{array}\]

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 j < -1.638603102240291e-07

    1. Initial program 5.5

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

    2. Initial simplification6.1

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

    4. Applied add-sqr-sqrt32.7

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

    5. Applied *-un-lft-identity32.7

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

    6. Applied prod-diff32.7

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

    7. Simplified32.5

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

    8. Simplified5.7

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

    if -1.638603102240291e-07 < j < 7.036950341320534e-163

    1. Initial program 5.0

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

    2. Initial simplification5.2

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

    4. Applied associate-*l*4.8

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

    if 7.036950341320534e-163 < j

    1. Initial program 6.0

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

    2. Initial simplification6.6

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

    3. Taylor expanded around -inf 6.5

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

    4. Simplified4.4

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

    6. Applied associate-*l*4.4

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

    8. Applied associate-*l*4.4

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;j \le -1.638603102240291 \cdot 10^{-07}:\\ \;\;\;\;(\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(t \cdot z\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* + \left(\left(j \cdot k\right) \cdot \left(-27.0\right) - i \cdot \left(4.0 \cdot x\right)\right)\\ \mathbf{elif}\;j \le 7.036950341320534 \cdot 10^{-163}:\\ \;\;\;\;(t \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot y\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(18.0 \cdot \left(x \cdot t\right)\right) \cdot z\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(x \cdot \left(4.0 \cdot i\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018252 +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)))