Average Error: 3.3 → 1.1
Time: 2.6m
Precision: 64
Internal Precision: 384
\[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -6.798242451867544 \cdot 10^{-115}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \mathbf{if}\;y \le 4.7969606968471655 \cdot 10^{-108}:\\ \;\;\;\;\left(27.0 \cdot \left(b \cdot a\right) + 2.0 \cdot x\right) - 9.0 \cdot \left(\left(z \cdot y\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

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

Original3.3
Target2.4
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + a \cdot \left(27.0 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2.0 - 9.0 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27.0\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -6.798242451867544e-115 or 4.7969606968471655e-108 < y

    1. Initial program 5.0

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-*l*1.4

      \[\leadsto \left(x \cdot 2.0 - \color{blue}{\left(y \cdot 9.0\right) \cdot \left(z \cdot t\right)}\right) + \left(a \cdot 27.0\right) \cdot b\]

    if -6.798242451867544e-115 < y < 4.7969606968471655e-108

    1. Initial program 0.6

      \[\left(x \cdot 2.0 - \left(\left(y \cdot 9.0\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27.0\right) \cdot b\]

    2. Taylor expanded around inf 0.5

      \[\leadsto \color{blue}{\left(27.0 \cdot \left(b \cdot a\right) + 2.0 \cdot x\right) - 9.0 \cdot \left(z \cdot \left(y \cdot t\right)\right)}\]
    3. Using strategy rm

    4. Applied associate-*r*0.6

      \[\leadsto \left(27.0 \cdot \left(b \cdot a\right) + 2.0 \cdot x\right) - 9.0 \cdot \color{blue}{\left(\left(z \cdot y\right) \cdot t\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 2.6m)Debug log

herbie shell --seed '#(1567391828 2030694642 2833800258 828025724 3004380912 3532991858)' +o setup:early-exit +o reduce:binary-search
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"

  :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)))