Average Error: 5.3 → 1.6
Time: 1.8m
Precision: 64
Internal Precision: 320
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -7.102389556456913 \cdot 10^{+68} \lor \neg \left(z \le 4.414924258486527 \cdot 10^{-24}\right):\\ \;\;\;\;(\left(-1\right) \cdot \left((x \cdot \left(i \cdot 4.0\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\right) + \left((\left(x \cdot \left(y \cdot \left(18.0 \cdot t\right)\right)\right) \cdot z + \left((\left(4.0 \cdot a\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(18.0 \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left((\left(4.0 \cdot a\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(x \cdot 4.0\right)\right))_*\\ \end{array}\]

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 2 regimes

  2. if z < -7.102389556456913e+68 or 4.414924258486527e-24 < z

    1. Initial program 7.4

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

    2. Initial simplification12.7

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

    4. Applied *-un-lft-identity12.7

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

    5. Applied *-un-lft-identity12.7

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

    6. Applied prod-diff12.7

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

    7. Simplified1.7

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

    8. Simplified1.7

      \[\leadsto (\left(-1\right) \cdot \left((x \cdot \left(4.0 \cdot i\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\right) + \left((\left(\left(18.0 \cdot t\right) \cdot \left(y \cdot x\right)\right) \cdot z + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_*\right))_* + \color{blue}{0}\]
    9. Using strategy rm

    10. Applied associate-*r*2.2

      \[\leadsto (\left(-1\right) \cdot \left((x \cdot \left(4.0 \cdot i\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\right) + \left((\color{blue}{\left(\left(\left(18.0 \cdot t\right) \cdot y\right) \cdot x\right)} \cdot z + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_*\right))_* + 0\]

    if -7.102389556456913e+68 < z < 4.414924258486527e-24

    1. Initial program 3.9

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

    2. Initial simplification1.2

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

    3. Taylor expanded around inf 1.2

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

    4. Simplified3.1

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

    5. Taylor expanded around -inf 1.2

      \[\leadsto (y \cdot \color{blue}{\left(18.0 \cdot \left(t \cdot \left(x \cdot z\right)\right)\right)} + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(\left(x \cdot 4.0\right) \cdot i\right))_*\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -7.102389556456913 \cdot 10^{+68} \lor \neg \left(z \le 4.414924258486527 \cdot 10^{-24}\right):\\ \;\;\;\;(\left(-1\right) \cdot \left((x \cdot \left(i \cdot 4.0\right) + \left(j \cdot \left(k \cdot 27.0\right)\right))_*\right) + \left((\left(x \cdot \left(y \cdot \left(18.0 \cdot t\right)\right)\right) \cdot z + \left((\left(4.0 \cdot a\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(18.0 \cdot \left(t \cdot \left(x \cdot z\right)\right)\right) + \left((\left(4.0 \cdot a\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(x \cdot 4.0\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018242 +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)))