Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 3.1

Time: 10.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.30679209408064958707058338211786093679 \cdot 10^{99}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;y \le 3.71994351214166424029575806263346983645 \cdot 10^{-65}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r564790 = x;
        double r564791 = y;
        double r564792 = r564790 * r564791;
        double r564793 = z;
        double r564794 = r564793 * r564791;
        double r564795 = r564792 - r564794;
        double r564796 = t;
        double r564797 = r564795 * r564796;
        return r564797;
}

↓

double f(double x, double y, double z, double t) {
        double r564798 = y;
        double r564799 = -1.3067920940806496e+99;
        bool r564800 = r564798 <= r564799;
        double r564801 = x;
        double r564802 = z;
        double r564803 = r564801 - r564802;
        double r564804 = t;
        double r564805 = r564803 * r564804;
        double r564806 = r564798 * r564805;
        double r564807 = 3.7199435121416642e-65;
        bool r564808 = r564798 <= r564807;
        double r564809 = r564798 * r564803;
        double r564810 = r564809 * r564804;
        double r564811 = r564804 * r564798;
        double r564812 = r564811 * r564803;
        double r564813 = r564808 ? r564810 : r564812;
        double r564814 = r564800 ? r564806 : r564813;
        return r564814;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	3.1
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;t \lt -9.231879582886776938073886590448747944753 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.543067051564877116200336808272775217995 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -1.3067920940806496e+99

   1. Initial program 21.6

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

   2. Simplified 21.6

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

   4. Applied associate-*l* 4.7

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

   if -1.3067920940806496e+99 < y < 3.7199435121416642e-65

   1. Initial program 2.6

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

   2. Simplified 2.6

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

   if 3.7199435121416642e-65 < y

   1. Initial program 12.1

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

   2. Simplified 12.1

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

   4. Applied associate-*l* 2.8

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

   6. Applied add-cube-cbrt 3.8

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

   8. Applied add-cube-cbrt 4.1

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

   9. Applied associate-*l* 4.1

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

   10. Simplified 3.8

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

   12. Applied pow1 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\left(x - z\right) \cdot t\right) \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right)\]

   13. Applied pow1 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\left(x - z\right) \cdot \color{blue}{{t}^{1}}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\]

   14. Applied pow1 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\color{blue}{{\left(x - z\right)}^{1}} \cdot {t}^{1}\right) \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\]

   15. Applied pow-prod-down 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\color{blue}{{\left(\left(x - z\right) \cdot t\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right)\]

   16. Applied pow-prod-down 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \color{blue}{{\left(\left(\left(x - z\right) \cdot t\right) \cdot \sqrt[3]{y}\right)}^{1}}\]

   17. Applied pow1 3.8

       \[\leadsto \left(\sqrt[3]{y} \cdot \color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}\right) \cdot {\left(\left(\left(x - z\right) \cdot t\right) \cdot \sqrt[3]{y}\right)}^{1}\]

   18. Applied pow1 3.8

       \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}} \cdot {\left(\sqrt[3]{y}\right)}^{1}\right) \cdot {\left(\left(\left(x - z\right) \cdot t\right) \cdot \sqrt[3]{y}\right)}^{1}\]

   19. Applied pow-prod-down 3.8

       \[\leadsto \color{blue}{{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)}^{1}} \cdot {\left(\left(\left(x - z\right) \cdot t\right) \cdot \sqrt[3]{y}\right)}^{1}\]

   20. Applied pow-prod-down 3.8

       \[\leadsto \color{blue}{{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(\left(\left(x - z\right) \cdot t\right) \cdot \sqrt[3]{y}\right)\right)}^{1}}\]

   21. Simplified 3.7

       \[\leadsto {\color{blue}{\left(\left(t \cdot y\right) \cdot \left(x - z\right)\right)}}^{1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.30679209408064958707058338211786093679 \cdot 10^{99}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;y \le 3.71994351214166424029575806263346983645 \cdot 10^{-65}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))