Average Error: 5.3 → 2.1
Time: 21.2s
Precision: 64
Internal Precision: 128
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.7091219761736728 \cdot 10^{+157}:\\ \;\;\;\;(y \cdot \left(\left(t \cdot x\right) \cdot \left(z \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(4.0 \cdot \left(i \cdot x\right)\right))_*\\ \mathbf{elif}\;y \le 6.61934784969459 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(\left(\left(z \cdot \left(y \cdot \left(x \cdot 18.0\right)\right)\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - i \cdot \left(4.0 \cdot x\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot t\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \end{array}\]

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 y < -2.7091219761736728e+157

    1. Initial program 14.6

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

    2. Simplified17.3

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

    4. Applied associate-*l*9.2

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

    5. Taylor expanded around -inf 17.1

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

    6. Simplified1.8

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

    7. Taylor expanded around -inf 1.8

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

    if -2.7091219761736728e+157 < y < 6.61934784969459e-78

    1. Initial program 2.1

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

    if 6.61934784969459e-78 < y

    1. Initial program 9.1

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

    2. Simplified9.8

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

    4. Applied associate-*l*5.7

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

    5. Taylor expanded around -inf 9.7

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

    6. Simplified2.7

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

    8. Applied associate-*r*2.3

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

    10. Applied associate-*l*2.3

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

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.7091219761736728 \cdot 10^{+157}:\\ \;\;\;\;(y \cdot \left(\left(t \cdot x\right) \cdot \left(z \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(4.0 \cdot \left(i \cdot x\right)\right))_*\\ \mathbf{elif}\;y \le 6.61934784969459 \cdot 10^{-78}:\\ \;\;\;\;\left(\left(\left(\left(z \cdot \left(y \cdot \left(x \cdot 18.0\right)\right)\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + c \cdot b\right) - i \cdot \left(4.0 \cdot x\right)\right) - \left(27.0 \cdot j\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;(y \cdot \left(\left(\left(x \cdot 18.0\right) \cdot z\right) \cdot t\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - (j \cdot \left(k \cdot 27.0\right) + \left(i \cdot \left(4.0 \cdot x\right)\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +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)))