Average Error: 5.4 → 4.5
Time: 53.3s
Precision: 64
Internal Precision: 384
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.88152462912295 \cdot 10^{-120}:\\ \;\;\;\;(\left(18.0 \cdot \left(\left(z \cdot y\right) \cdot x\right) - 4.0 \cdot a\right) \cdot t + \left((\left(x \cdot 4.0\right) \cdot \left(-i\right) + \left(b \cdot c\right))_* - \left(27.0 \cdot k\right) \cdot j\right))_*\\ \mathbf{if}\;t \le 1.0711434725909347 \cdot 10^{-204}:\\ \;\;\;\;(\left(\left(-a\right) \cdot 4.0\right) \cdot t + \left((\left(x \cdot 4.0\right) \cdot \left(-i\right) + \left(c \cdot b\right))_* - k \cdot \left(27.0 \cdot j\right)\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\left(z \cdot x\right) \cdot \left(y \cdot 18.0\right) - 4.0 \cdot a\right) \cdot t + \left((\left(x \cdot 4.0\right) \cdot \left(-i\right) + \left(b \cdot c\right))_* - 27.0 \cdot \left(k \cdot j\right)\right))_*\\ \end{array}\]

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 3 regimes

  2. if t < -3.88152462912295e-120

    1. Initial program 3.2

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

    2. Applied simplify3.3

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

    3. Taylor expanded around inf 3.2

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

    5. Applied associate-*r*3.5

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

    if -3.88152462912295e-120 < t < 1.0711434725909347e-204

    1. Initial program 8.6

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

    2. Applied simplify8.6

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

    3. Taylor expanded around 0 6.0

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

    4. Applied simplify6.1

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

    if 1.0711434725909347e-204 < t

    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 simplify4.4

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

    4. Applied associate-*l*4.2

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

Runtime

Time bar (total: 53.3s)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)))