Average Error: 5.4 → 3.3
Time: 28.8s
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.6466238528257665 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(\left((\left(-t \cdot 4.0\right) \cdot a + \left(\left(t \cdot 4.0\right) \cdot a\right))_* + b \cdot c\right) + (\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(t \cdot z\right) + \left(\left(-a\right) \cdot \left(t \cdot 4.0\right)\right))_*\right) - i \cdot \left(4.0 \cdot x\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.995025664542542 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) - \left(t \cdot 4.0\right) \cdot a\right)\right) - i \cdot \left(4.0 \cdot x\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(27.0 \cdot k\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \end{array}\]

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 < -2.6466238528257665e-14

    1. Initial program 1.5

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

    3. Applied associate-*l*7.5

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

    5. Applied associate-*l*7.6

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

    7. Applied associate-*l*7.7

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

    9. Applied prod-diff7.7

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

    10. Applied associate-+l+7.7

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

    if -2.6466238528257665e-14 < t < 2.995025664542542e-122

    1. Initial program 8.4

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

    3. Applied associate-*l*4.3

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

    5. Applied associate-*l*4.3

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

    7. Applied associate-*l*4.3

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

    9. Applied associate-*l*1.4

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

    if 2.995025664542542e-122 < t

    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. Initial simplification3.5

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.6466238528257665 \cdot 10^{-14}:\\ \;\;\;\;\left(\left(\left((\left(-t \cdot 4.0\right) \cdot a + \left(\left(t \cdot 4.0\right) \cdot a\right))_* + b \cdot c\right) + (\left(x \cdot \left(18.0 \cdot y\right)\right) \cdot \left(t \cdot z\right) + \left(\left(-a\right) \cdot \left(t \cdot 4.0\right)\right))_*\right) - i \cdot \left(4.0 \cdot x\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{elif}\;t \le 2.995025664542542 \cdot 10^{-122}:\\ \;\;\;\;\left(\left(b \cdot c + \left(x \cdot \left(\left(t \cdot z\right) \cdot \left(18.0 \cdot y\right)\right) - \left(t \cdot 4.0\right) \cdot a\right)\right) - i \cdot \left(4.0 \cdot x\right)\right) - k \cdot \left(j \cdot 27.0\right)\\ \mathbf{else}:\\ \;\;\;\;(t \cdot \left(\left(z \cdot y\right) \cdot \left(x \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(b \cdot c\right))_*\right))_* - (j \cdot \left(27.0 \cdot k\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \end{array}\]

Runtime

Time bar (total: 28.8s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes5.63.30.35.243.1%
herbie shell --seed 2018296 +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)))