Average Error: 5.4 → 2.8
Time: 1.3m
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.855484736673346 \cdot 10^{+146}:\\ \;\;\;\;\left(\left(\left(\left(\sqrt[3]{\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t} \cdot \sqrt[3]{\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t}\right) \cdot \sqrt[3]{\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t} - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{if}\;t \le 1.1790309122498075 \cdot 10^{-176}:\\ \;\;\;\;\left(\left(z \cdot 18.0\right) \cdot \left(\left(y \cdot t\right) \cdot x\right) + b \cdot c\right) - (4.0 \cdot \left((a \cdot t + \left(x \cdot i\right))_*\right) + \left(\left(27.0 \cdot k\right) \cdot j\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(z \cdot x\right) \cdot \left(y \cdot 18.0\right)\right) \cdot t + \left(b \cdot c\right))_* - (4.0 \cdot \left((t \cdot a + \left(x \cdot i\right))_*\right) + \left(\left(j \cdot k\right) \cdot 27.0\right))_*\\ \end{array}\]

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 t < -1.855484736673346e+146

    1. Initial program 1.1

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

    3. Applied add-cube-cbrt1.3

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

    if -1.855484736673346e+146 < t < 1.1790309122498075e-176

    1. Initial program 6.7

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

    2. Taylor expanded around inf 2.5

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

    3. Applied simplify2.5

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

    5. Applied associate-*r*2.4

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

    7. Applied fma-udef2.4

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

    if 1.1790309122498075e-176 < t

    1. Initial program 4.2

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

    2. Taylor expanded around 0 4.1

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

    3. Applied simplify3.6

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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)))