Average Error: 5.3 → 2.5
Time: 9.4m
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.0328664264298304 \cdot 10^{-192}:\\ \;\;\;\;\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(18.0 \cdot \left(z \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)\\ \mathbf{if}\;z \le 2.001548979293268 \cdot 10^{-116}:\\ \;\;\;\;\left(\sqrt[3]{\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)} \cdot \sqrt[3]{\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)}\right) \cdot \sqrt[3]{\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(t \cdot \left(\left(18.0 \cdot x\right) \cdot \left(y \cdot z\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot c - 4.0 \cdot \left(x \cdot i + a \cdot t\right)\right) + \left(18.0 \cdot \left(z \cdot \left(y \cdot \left(t \cdot x\right)\right)\right) - 27.0 \cdot \left(k \cdot j\right)\right)\\ \end{array}\]

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 2 regimes

  2. if z < -1.0328664264298304e-192 or 2.001548979293268e-116 < 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 simplify8.0

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

    3. Taylor expanded around inf 2.8

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

    if -1.0328664264298304e-192 < z < 2.001548979293268e-116

    1. Initial program 4.7

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

    2. Applied simplify0.5

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

    4. Applied add-cube-cbrt1.7

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

Runtime

Time bar (total: 9.4m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))