Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 2.5

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6648838448.446291 \lor \neg \left(t \le 6.1434809147908996 \cdot 10^{33}\right):\\ \;\;\;\;t \cdot \left(x \cdot y\right) + \left(-t \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot \left(x - z\right)\right) \cdot y\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r600951 = x;
        double r600952 = y;
        double r600953 = r600951 * r600952;
        double r600954 = z;
        double r600955 = r600954 * r600952;
        double r600956 = r600953 - r600955;
        double r600957 = t;
        double r600958 = r600956 * r600957;
        return r600958;
}

↓

double f(double x, double y, double z, double t) {
        double r600959 = t;
        double r600960 = -6648838448.446291;
        bool r600961 = r600959 <= r600960;
        double r600962 = 6.1434809147909e+33;
        bool r600963 = r600959 <= r600962;
        double r600964 = !r600963;
        bool r600965 = r600961 || r600964;
        double r600966 = x;
        double r600967 = y;
        double r600968 = r600966 * r600967;
        double r600969 = r600959 * r600968;
        double r600970 = z;
        double r600971 = r600970 * r600967;
        double r600972 = r600959 * r600971;
        double r600973 = -r600972;
        double r600974 = r600969 + r600973;
        double r600975 = r600966 - r600970;
        double r600976 = r600959 * r600975;
        double r600977 = r600976 * r600967;
        double r600978 = 1.0;
        double r600979 = pow(r600977, r600978);
        double r600980 = r600965 ? r600974 : r600979;
        return r600980;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	3.2
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if t < -6648838448.446291 or 6.1434809147909e+33 < t

   1. Initial program 3.4

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

   2. Simplified 3.4

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

   4. Applied add-cube-cbrt 4.5

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

   5. Applied associate-*l* 4.5

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

   7. Applied sub-neg 4.5

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

   8. Applied distribute-lft-in 4.5

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

   9. Applied distribute-lft-in 4.5

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

   10. Applied distribute-lft-in 4.5

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

   11. Simplified 4.0

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

   12. Simplified 3.4

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

   if -6648838448.446291 < t < 6.1434809147909e+33

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   4. Applied add-cube-cbrt 9.8

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

   5. Applied associate-*l* 9.8

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

   7. Applied pow1 9.8

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

   8. Applied pow1 9.8

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

   9. Applied pow-prod-down 9.8

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

   10. Applied pow1 9.8

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

   11. Applied pow1 9.8

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

   12. Applied pow-prod-down 9.8

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

   13. Applied pow-prod-down 9.8

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

   14. Applied pow1 9.8

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

   15. Applied pow-prod-down 9.8

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

   16. Simplified 2.0

       \[\leadsto {\color{blue}{\left(\left(t \cdot \left(x - z\right)\right) \cdot y\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6648838448.446291 \lor \neg \left(t \le 6.1434809147908996 \cdot 10^{33}\right):\\ \;\;\;\;t \cdot \left(x \cdot y\right) + \left(-t \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(t \cdot \left(x - z\right)\right) \cdot y\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))