Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.8 → 1.2

Time: 2.3s

Precision: 64

Math

\[\left(x \cdot y - z \cdot y\right) \cdot t\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y = -inf.0:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -1.6211301324052328 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.12671917784 \cdot 10^{-313}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.40327700031230854 \cdot 10^{82}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	3.0
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;t \lt -9.2318795828867769 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.5430670515648771 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -inf.0

   1. Initial program 64.0

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(x - z\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 0.3

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(x - z\right)}\]
   5. Using strategy rm

   6. Applied sub-neg 0.3

      \[\leadsto \left(t \cdot y\right) \cdot \color{blue}{\left(x + \left(-z\right)\right)}\]

   7. Applied distribute-lft-in 0.3

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)}\]

   if -inf.0 < (- (* x y) (* z y)) < -1.6211301324052328e-137 or 1.1267191778431e-313 < (- (* x y) (* z y)) < 1.4032770003123085e+82

   1. Initial program 0.2

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]

   if -1.6211301324052328e-137 < (- (* x y) (* z y)) < 1.1267191778431e-313 or 1.4032770003123085e+82 < (- (* x y) (* z y))

   1. Initial program 12.5

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]
   2. Using strategy rm

   3. Applied distribute-rgt-out-- 12.5

      \[\leadsto \color{blue}{\left(y \cdot \left(x - z\right)\right)} \cdot t\]

   4. Applied associate-*l* 3.1

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y = -inf.0:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -1.6211301324052328 \cdot 10^{-137}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.12671917784 \cdot 10^{-313}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.40327700031230854 \cdot 10^{82}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))