Average Error: 5.3 → 1.6
Time: 46.8s
Precision: 64
Internal Precision: 576
\[\left(\left(\left(\left(\left(\left(x \cdot 18.0\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4.0\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4.0\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.4251161474381644 \cdot 10^{-67}:\\ \;\;\;\;\left((y \cdot \left(\left(t \cdot x\right) \cdot \left(z \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - \left(4.0 \cdot x\right) \cdot i\right) - \sqrt[3]{27.0 \cdot \left(j \cdot k\right)} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}}\right) \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)}\right)\\ \mathbf{elif}\;y \le 7.887983587283119 \cdot 10^{+47}:\\ \;\;\;\;\left(\left(c \cdot b + \left(18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - 4.0 \cdot \left(t \cdot a\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left((y \cdot \left(18.0 \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

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 < -1.4251161474381644e-67

    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. Taylor expanded around inf 9.3

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

    3. Simplified2.2

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

    4. Taylor expanded around 0 2.2

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

    6. Applied add-cube-cbrt2.4

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

    8. Applied add-cube-cbrt2.5

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

    if -1.4251161474381644e-67 < y < 7.887983587283119e+47

    1. Initial program 1.2

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

    2. Taylor expanded around -inf 1.1

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

    if 7.887983587283119e+47 < y

    1. Initial program 12.3

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

    2. Taylor expanded around inf 13.4

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

    3. Simplified2.0

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

    4. Taylor expanded around 0 1.9

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

    5. Taylor expanded around -inf 1.5

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.4251161474381644 \cdot 10^{-67}:\\ \;\;\;\;\left((y \cdot \left(\left(t \cdot x\right) \cdot \left(z \cdot 18.0\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - \left(4.0 \cdot x\right) \cdot i\right) - \sqrt[3]{27.0 \cdot \left(j \cdot k\right)} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}} \cdot \sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{27.0 \cdot \left(j \cdot k\right)}}\right) \cdot \sqrt[3]{27.0 \cdot \left(j \cdot k\right)}\right)\\ \mathbf{elif}\;y \le 7.887983587283119 \cdot 10^{+47}:\\ \;\;\;\;\left(\left(c \cdot b + \left(18.0 \cdot \left(t \cdot \left(x \cdot \left(z \cdot y\right)\right)\right) - 4.0 \cdot \left(t \cdot a\right)\right)\right) - \left(4.0 \cdot x\right) \cdot i\right) - \left(j \cdot 27.0\right) \cdot k\\ \mathbf{else}:\\ \;\;\;\;\left((y \cdot \left(18.0 \cdot \left(\left(z \cdot x\right) \cdot t\right)\right) + \left((\left(a \cdot 4.0\right) \cdot \left(-t\right) + \left(c \cdot b\right))_*\right))_* - \left(4.0 \cdot x\right) \cdot i\right) - 27.0 \cdot \left(j \cdot k\right)\\ \end{array}\]

Runtime

Time bar (total: 46.8s)Debug logProfile

herbie shell --seed 2018234 +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)))