Average Error: 5.7 → 1.8
Time: 2.7m
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.922874448615507 \cdot 10^{+75}:\\ \;\;\;\;\left((\left(\left(t \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) \cdot z + \left(b \cdot c\right))_* - 4.0 \cdot \left(i \cdot x + a \cdot t\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{if}\;z \le 1.7801511596592231 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) \cdot t - t \cdot \left(4.0 \cdot a\right)\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z + \left((\left(-a\right) \cdot \left(t \cdot 4.0\right) + \left(b \cdot c\right))_*\right))_* - (i \cdot \left(x \cdot 4.0\right) + \left(j \cdot \left(27.0 \cdot k\right)\right))_*\\ \end{array}\]

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 z < -3.922874448615507e+75

    1. Initial program 8.1

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

    3. Applied *-un-lft-identity8.1

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

    4. Applied prod-diff8.1

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

    5. Applied simplify1.5

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

    6. Applied simplify1.5

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

    if -3.922874448615507e+75 < z < 1.7801511596592231e+22

    1. Initial program 4.5

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

    3. Applied associate-*l*1.8

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

    if 1.7801511596592231e+22 < z

    1. Initial program 7.6

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

    2. Applied simplify2.2

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

  4. Applied simplify1.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;z \le -3.922874448615507 \cdot 10^{+75}:\\ \;\;\;\;\left((\left(\left(t \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) \cdot z + \left(b \cdot c\right))_* - 4.0 \cdot \left(i \cdot x + a \cdot t\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{if}\;z \le 1.7801511596592231 \cdot 10^{+22}:\\ \;\;\;\;\left(\left(b \cdot c + \left(\left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) \cdot t - t \cdot \left(4.0 \cdot a\right)\right)\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(t \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot z + \left((\left(-a\right) \cdot \left(t \cdot 4.0\right) + \left(b \cdot c\right))_*\right))_* - (i \cdot \left(x \cdot 4.0\right) + \left(j \cdot \left(27.0 \cdot k\right)\right))_*\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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)))