Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 0.5

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -6.9057663882231999 \cdot 10^{281}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -1.0767303544454203 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.71850630441090114 \cdot 10^{-140}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.47858745386993944 \cdot 10^{290}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	3.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -6.9057663882231999e281

   1. Initial program 50.8

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

   3. Applied distribute-rgt-out-- 50.8

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

   4. Applied associate-*l* 0.3

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

   if -6.9057663882231999e281 < (- (* x y) (* z y)) < -1.0767303544454203e-139 or 6.71850630441090114e-140 < (- (* x y) (* z y)) < 1.47858745386993944e290

   1. Initial program 0.3

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

   if -1.0767303544454203e-139 < (- (* x y) (* z y)) < 6.71850630441090114e-140 or 1.47858745386993944e290 < (- (* x y) (* z y))

   1. Initial program 16.4

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

   2. Simplified 16.4

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

   4. Applied associate-*r* 1.3

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

   6. Applied sub-neg 1.3

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

   7. Applied distribute-lft-in 1.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -6.9057663882231999 \cdot 10^{281}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -1.0767303544454203 \cdot 10^{-139}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.71850630441090114 \cdot 10^{-140}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.47858745386993944 \cdot 10^{290}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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))