Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, A

Average Error: 3.6 → 1.6

Time: 10.6s

Precision: binary64

Math

\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -3.53682367306220511 \cdot 10^{-187} \lor \neg \left(t \le 5.76849963678283904 \cdot 10^{90}\right):\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(y \cdot \left(9 \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) + y \cdot \left(t \cdot \left(9 \cdot \left(-z\right)\right)\right)\right)\\ \end{array}\]

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

Target

Original	3.6
Target	2.7
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -3.53682367306220511e-187 or 5.76849963678283904e90 < t

   1. Initial program 1.9

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 1.9

      \[\leadsto \left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot \color{blue}{\left(1 \cdot t\right)}\right) + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-*r* 1.9

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(\left(\left(y \cdot 9\right) \cdot z\right) \cdot 1\right) \cdot t}\right) + \left(a \cdot 27\right) \cdot b\]

   5. Simplified 2.0

      \[\leadsto \left(x \cdot 2 - \color{blue}{\left(y \cdot \left(9 \cdot z\right)\right)} \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

   if -3.53682367306220511e-187 < t < 5.76849963678283904e90

   1. Initial program 5.4

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
   2. Using strategy rm

   3. Applied sub-neg 5.4

      \[\leadsto \color{blue}{\left(x \cdot 2 + \left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\right)} + \left(a \cdot 27\right) \cdot b\]

   4. Applied associate-+l+ 5.4

      \[\leadsto \color{blue}{x \cdot 2 + \left(\left(-\left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\right)}\]

   5. Simplified 1.3

      \[\leadsto x \cdot 2 + \color{blue}{\left(a \cdot \left(27 \cdot b\right) + y \cdot \left(\left(9 \cdot z\right) \cdot \left(-t\right)\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.53682367306220511 \cdot 10^{-187} \lor \neg \left(t \le 5.76849963678283904 \cdot 10^{90}\right):\\ \;\;\;\;\left(x \cdot 2 - t \cdot \left(y \cdot \left(9 \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;x \cdot 2 + \left(a \cdot \left(27 \cdot b\right) + y \cdot \left(t \cdot \left(9 \cdot \left(-z\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))