Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.0 → 3.1

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.7183563329919357 \cdot 10^{77}:\\ \;\;\;\;{\left(\frac{1}{\frac{y - t}{2}} \cdot \frac{x}{z}\right)}^{1}\\ \mathbf{elif}\;z \le 8.1507283132915605 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{\frac{\left(y - t\right) \cdot z}{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{z}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r644588 = x;
        double r644589 = 2.0;
        double r644590 = r644588 * r644589;
        double r644591 = y;
        double r644592 = z;
        double r644593 = r644591 * r644592;
        double r644594 = t;
        double r644595 = r644594 * r644592;
        double r644596 = r644593 - r644595;
        double r644597 = r644590 / r644596;
        return r644597;
}

↓

double f(double x, double y, double z, double t) {
        double r644598 = z;
        double r644599 = -2.7183563329919357e+77;
        bool r644600 = r644598 <= r644599;
        double r644601 = 1.0;
        double r644602 = y;
        double r644603 = t;
        double r644604 = r644602 - r644603;
        double r644605 = 2.0;
        double r644606 = r644604 / r644605;
        double r644607 = r644601 / r644606;
        double r644608 = x;
        double r644609 = r644608 / r644598;
        double r644610 = r644607 * r644609;
        double r644611 = pow(r644610, r644601);
        double r644612 = 8.15072831329156e-182;
        bool r644613 = r644598 <= r644612;
        double r644614 = r644604 * r644598;
        double r644615 = r644614 / r644605;
        double r644616 = r644608 / r644615;
        double r644617 = r644606 / r644608;
        double r644618 = r644601 / r644617;
        double r644619 = r644618 / r644598;
        double r644620 = pow(r644619, r644601);
        double r644621 = r644613 ? r644616 : r644620;
        double r644622 = r644600 ? r644611 : r644621;
        return r644622;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	2.1
Herbie	3.1

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -2.7183563329919357e+77

   1. Initial program 12.9

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

   2. Simplified 10.5

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

   4. Applied *-un-lft-identity 10.5

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

   5. Applied times-frac 10.5

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

   6. Applied *-un-lft-identity 10.5

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

   7. Applied times-frac 2.1

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

   8. Simplified 2.1

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
   9. Using strategy rm

   10. Applied pow1 2.1

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

   11. Applied pow1 2.1

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

   12. Applied pow-prod-down 2.1

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

   13. Simplified 2.0

       \[\leadsto {\color{blue}{\left(\frac{\frac{x}{\frac{y - t}{2}}}{z}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied clear-num 2.1

       \[\leadsto {\left(\frac{\color{blue}{\frac{1}{\frac{\frac{y - t}{2}}{x}}}}{z}\right)}^{1}\]
   16. Using strategy rm

   17. Applied *-un-lft-identity 2.1

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

   18. Applied div-inv 2.1

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

   19. Applied add-sqr-sqrt 2.1

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

   20. Applied times-frac 2.1

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

   21. Applied times-frac 2.1

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

   22. Simplified 2.1

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

   23. Simplified 2.1

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

   if -2.7183563329919357e+77 < z < 8.15072831329156e-182

   1. Initial program 3.1

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

   2. Simplified 3.0

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

   4. Applied *-commutative 3.0

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

   if 8.15072831329156e-182 < z

   1. Initial program 7.6

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

   2. Simplified 5.9

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

   4. Applied *-un-lft-identity 5.9

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

   5. Applied times-frac 5.9

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

   6. Applied *-un-lft-identity 5.9

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

   7. Applied times-frac 3.4

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

   8. Simplified 3.4

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{x}{\frac{y - t}{2}}\]
   9. Using strategy rm

   10. Applied pow1 3.4

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

   11. Applied pow1 3.4

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

   12. Applied pow-prod-down 3.4

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

   13. Simplified 3.4

       \[\leadsto {\color{blue}{\left(\frac{\frac{x}{\frac{y - t}{2}}}{z}\right)}}^{1}\]
   14. Using strategy rm

   15. Applied clear-num 3.6

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.7183563329919357 \cdot 10^{77}:\\ \;\;\;\;{\left(\frac{1}{\frac{y - t}{2}} \cdot \frac{x}{z}\right)}^{1}\\ \mathbf{elif}\;z \le 8.1507283132915605 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{\frac{\left(y - t\right) \cdot z}{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{1}{\frac{\frac{y - t}{2}}{x}}}{z}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2)))

  (/ (* x 2) (- (* y z) (* t z))))