Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 6.8 → 2.3

Time: 9.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -6.233745507656147839026965742862561565412 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{z}}{y - t}\\ \mathbf{elif}\;z \le 2629606188374439057894428064142589952:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \frac{2 \cdot x}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{z}}}{\sqrt[3]{y - t}}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r434478 = x;
        double r434479 = 2.0;
        double r434480 = r434478 * r434479;
        double r434481 = y;
        double r434482 = z;
        double r434483 = r434481 * r434482;
        double r434484 = t;
        double r434485 = r434484 * r434482;
        double r434486 = r434483 - r434485;
        double r434487 = r434480 / r434486;
        return r434487;
}

↓

double f(double x, double y, double z, double t) {
        double r434488 = z;
        double r434489 = -6.233745507656148e+81;
        bool r434490 = r434488 <= r434489;
        double r434491 = 2.0;
        double r434492 = x;
        double r434493 = r434491 * r434492;
        double r434494 = r434493 / r434488;
        double r434495 = y;
        double r434496 = t;
        double r434497 = r434495 - r434496;
        double r434498 = r434494 / r434497;
        double r434499 = 2.629606188374439e+36;
        bool r434500 = r434488 <= r434499;
        double r434501 = r434492 * r434491;
        double r434502 = r434488 * r434497;
        double r434503 = r434501 / r434502;
        double r434504 = 1.0;
        double r434505 = r434504 / r434488;
        double r434506 = cbrt(r434505);
        double r434507 = r434506 * r434506;
        double r434508 = cbrt(r434497);
        double r434509 = r434508 * r434508;
        double r434510 = r434493 / r434509;
        double r434511 = r434507 * r434510;
        double r434512 = cbrt(r434504);
        double r434513 = r434512 / r434488;
        double r434514 = cbrt(r434513);
        double r434515 = r434514 / r434508;
        double r434516 = r434511 * r434515;
        double r434517 = r434500 ? r434503 : r434516;
        double r434518 = r434490 ? r434498 : r434517;
        return r434518;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.8
Target	2.0
Herbie	2.3

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061113708240820439530037456 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.045027827330125861587720199944080049996 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -6.233745507656148e+81

   1. Initial program 12.7

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

   2. Simplified 9.8

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

   4. Applied div-inv 9.8

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

   6. Applied associate-/r* 9.4

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

   8. Applied pow1 9.4

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

   9. Applied pow1 9.4

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

   10. Applied pow1 9.4

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

   11. Applied pow-prod-down 9.4

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

   12. Applied pow-prod-down 9.4

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

   13. Simplified 2.0

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

   if -6.233745507656148e+81 < z < 2.629606188374439e+36

   1. Initial program 2.4

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

   2. Simplified 2.4

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

   if 2.629606188374439e+36 < z

   1. Initial program 12.0

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

   2. Simplified 9.8

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

   4. Applied div-inv 9.9

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

   6. Applied associate-/r* 9.3

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

   8. Applied add-cube-cbrt 9.7

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

   9. Applied *-un-lft-identity 9.7

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

   10. Applied add-cube-cbrt 9.7

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

   11. Applied times-frac 9.7

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

   12. Applied times-frac 9.7

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

   13. Applied associate-*r* 3.6

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

   14. Simplified 3.6

       \[\leadsto \color{blue}{\frac{2 \cdot x}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}} \cdot \frac{\frac{\sqrt[3]{1}}{z}}{\sqrt[3]{y - t}}\]
   15. Using strategy rm

   16. Applied *-un-lft-identity 3.6

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

   17. Applied cbrt-prod 3.6

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

   18. Applied add-cube-cbrt 3.8

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

   19. Applied times-frac 3.8

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

   20. Applied associate-*r* 2.1

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

   21. Simplified 2.1

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.233745507656147839026965742862561565412 \cdot 10^{81}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{z}}{y - t}\\ \mathbf{elif}\;z \le 2629606188374439057894428064142589952:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \frac{2 \cdot x}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{z}}}{\sqrt[3]{y - t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))))