Average Error: 5.1 → 2.2
Time: 7.7m
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1917768530699547 \cdot 10^{-115}:\\ \;\;\;\;\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(\left(t \cdot \left(\left(18.0 \cdot x\right) \cdot y\right)\right) \cdot z - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{if}\;z \le 1.8701868188734794 \cdot 10^{-140}:\\ \;\;\;\;\left(b \cdot c - 4.0 \cdot \left(\left(\sqrt[3]{x \cdot i + a \cdot t} \cdot \sqrt[3]{x \cdot i + a \cdot t}\right) \cdot \sqrt[3]{x \cdot i + a \cdot t}\right)\right) + \left(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(\left(18.0 \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) \cdot z - 27.0 \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

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 < -1.1917768530699547e-115

    1. Initial program 5.2

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

    2. Applied simplify8.5

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

    4. Applied associate-*r*5.1

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

    6. Applied associate-*r*2.5

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

    if -1.1917768530699547e-115 < z < 1.8701868188734794e-140

    1. Initial program 4.6

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

    2. Applied simplify0.4

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

    4. Applied add-cube-cbrt0.9

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

    if 1.8701868188734794e-140 < z

    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. Applied simplify7.9

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

    4. Applied associate-*r*5.4

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

    6. Applied associate-*r*3.1

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

    7. Taylor expanded around inf 3.1

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

Runtime

Time bar (total: 7.7m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(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)))