Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.2 → 5.1

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.6472230079801522 \cdot 10^{-139}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + \left(t \cdot \left(-z\right)\right) \cdot y\\ \mathbf{elif}\;y \le 2.9072852134162777 \cdot 10^{-123}:\\ \;\;\;\;\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(y \cdot \left(x - z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	3.3
Herbie	5.1

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

Derivation

1. Split input into 3 regimes

2. if y < -2.6472230079801522e-139

   1. Initial program 9.7

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

   2. Simplified 9.7

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

   4. Applied sub-neg 9.7

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

   5. Applied distribute-lft-in 9.7

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

   6. Applied distribute-lft-in 9.7

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

   7. Simplified 9.7

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

   8. Simplified 9.7

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

   10. Applied associate-*r* 7.9

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

   if -2.6472230079801522e-139 < y < 2.9072852134162777e-123

   1. Initial program 3.0

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

   2. Simplified 3.0

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

   4. Applied add-cube-cbrt 3.8

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)} \cdot \left(y \cdot \left(x - z\right)\right)\]

   5. Applied associate-*l* 3.8

      \[\leadsto \color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(y \cdot \left(x - z\right)\right)\right)}\]

   if 2.9072852134162777e-123 < y

   1. Initial program 9.8

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

   2. Simplified 9.8

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

   4. Applied associate-*r* 3.9

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

   6. Applied sub-neg 3.9

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

   7. Applied distribute-lft-in 4.0

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.6472230079801522 \cdot 10^{-139}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + \left(t \cdot \left(-z\right)\right) \cdot y\\ \mathbf{elif}\;y \le 2.9072852134162777 \cdot 10^{-123}:\\ \;\;\;\;\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \left(\sqrt[3]{t} \cdot \left(y \cdot \left(x - z\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020173 
(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))