Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.7 → 2.3

Time: 16.8s

Precision: 64

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -879582645775952851454505447842129578557400:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;z \le 5.968128368805632124167463283042718481054 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r24954878 = x;
        double r24954879 = 2.0;
        double r24954880 = r24954878 * r24954879;
        double r24954881 = y;
        double r24954882 = z;
        double r24954883 = r24954881 * r24954882;
        double r24954884 = t;
        double r24954885 = r24954884 * r24954882;
        double r24954886 = r24954883 - r24954885;
        double r24954887 = r24954880 / r24954886;
        return r24954887;
}

↓

double f(double x, double y, double z, double t) {
        double r24954888 = z;
        double r24954889 = -8.795826457759529e+41;
        bool r24954890 = r24954888 <= r24954889;
        double r24954891 = x;
        double r24954892 = 2.0;
        double r24954893 = r24954891 * r24954892;
        double r24954894 = r24954893 / r24954888;
        double r24954895 = y;
        double r24954896 = t;
        double r24954897 = r24954895 - r24954896;
        double r24954898 = r24954894 / r24954897;
        double r24954899 = 5.968128368805632e-47;
        bool r24954900 = r24954888 <= r24954899;
        double r24954901 = r24954897 * r24954888;
        double r24954902 = r24954893 / r24954901;
        double r24954903 = r24954900 ? r24954902 : r24954898;
        double r24954904 = r24954890 ? r24954898 : r24954903;
        return r24954904;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.7
Target	2.2
Herbie	2.3

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330126029709547581125571222799 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -8.795826457759529e+41 or 5.968128368805632e-47 < z

   1. Initial program 10.2

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 2.1

      \[\leadsto \color{blue}{\frac{2}{y - t} \cdot \frac{x}{z}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 2.6

      \[\leadsto \frac{2}{y - t} \cdot \frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity 2.6

      \[\leadsto \frac{2}{y - t} \cdot \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac 2.6

      \[\leadsto \frac{2}{y - t} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r* 5.3

      \[\leadsto \color{blue}{\left(\frac{2}{y - t} \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{x}{\sqrt[3]{z}}}\]

   8. Simplified 5.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{y - t}} \cdot \frac{x}{\sqrt[3]{z}}\]
   9. Using strategy rm

   10. Applied associate-*l/ 2.6

       \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z}}}{y - t}}\]

   11. Simplified 2.0

       \[\leadsto \frac{\color{blue}{\frac{x \cdot 2}{z}}}{y - t}\]

   if -8.795826457759529e+41 < z < 5.968128368805632e-47

   1. Initial program 2.7

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 10.5

      \[\leadsto \color{blue}{\frac{2}{y - t} \cdot \frac{x}{z}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 11.4

      \[\leadsto \frac{2}{y - t} \cdot \frac{x}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity 11.4

      \[\leadsto \frac{2}{y - t} \cdot \frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac 11.4

      \[\leadsto \frac{2}{y - t} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r* 4.6

      \[\leadsto \color{blue}{\left(\frac{2}{y - t} \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{x}{\sqrt[3]{z}}}\]

   8. Simplified 4.6

      \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{y - t}} \cdot \frac{x}{\sqrt[3]{z}}\]
   9. Using strategy rm

   10. Applied associate-*l/ 11.4

       \[\leadsto \color{blue}{\frac{\frac{2}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\sqrt[3]{z}}}{y - t}}\]

   11. Simplified 10.5

       \[\leadsto \frac{\color{blue}{\frac{x \cdot 2}{z}}}{y - t}\]
   12. Using strategy rm

   13. Applied div-inv 10.6

       \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right) \cdot \frac{1}{z}}}{y - t}\]

   14. Applied associate-/l* 2.8

       \[\leadsto \color{blue}{\frac{x \cdot 2}{\frac{y - t}{\frac{1}{z}}}}\]

   15. Simplified 2.7

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -879582645775952851454505447842129578557400:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \mathbf{elif}\;z \le 5.968128368805632124167463283042718481054 \cdot 10^{-47}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))