Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.3 → 2.4

Time: 8.0s

Precision: 64

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

Math

\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -237562455431070744576 \lor \neg \left(\frac{1}{x} \le 2.344126945452436231818170630862504221596 \cdot 10^{59}\right):\\ \;\;\;\;\frac{\sqrt[3]{1}}{\left(\left(x \cdot z\right) \cdot z + 1 \cdot x\right) \cdot y} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}}{{z}^{2} + 1} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{1}}}{y}}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r259436 = 1.0;
        double r259437 = x;
        double r259438 = r259436 / r259437;
        double r259439 = y;
        double r259440 = z;
        double r259441 = r259440 * r259440;
        double r259442 = r259436 + r259441;
        double r259443 = r259439 * r259442;
        double r259444 = r259438 / r259443;
        return r259444;
}

↓

double f(double x, double y, double z) {
        double r259445 = 1.0;
        double r259446 = x;
        double r259447 = r259445 / r259446;
        double r259448 = -2.3756245543107074e+20;
        bool r259449 = r259447 <= r259448;
        double r259450 = 2.3441269454524362e+59;
        bool r259451 = r259447 <= r259450;
        double r259452 = !r259451;
        bool r259453 = r259449 || r259452;
        double r259454 = cbrt(r259445);
        double r259455 = z;
        double r259456 = r259446 * r259455;
        double r259457 = r259456 * r259455;
        double r259458 = r259445 * r259446;
        double r259459 = r259457 + r259458;
        double r259460 = y;
        double r259461 = r259459 * r259460;
        double r259462 = r259454 / r259461;
        double r259463 = r259454 * r259454;
        double r259464 = r259462 * r259463;
        double r259465 = cbrt(r259454);
        double r259466 = r259465 * r259465;
        double r259467 = 2.0;
        double r259468 = pow(r259455, r259467);
        double r259469 = r259468 + r259445;
        double r259470 = r259466 / r259469;
        double r259471 = r259465 / r259460;
        double r259472 = r259471 / r259446;
        double r259473 = r259470 * r259472;
        double r259474 = r259463 * r259473;
        double r259475 = r259453 ? r259464 : r259474;
        return r259475;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	5.6
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 x) < -2.3756245543107074e+20 or 2.3441269454524362e+59 < (/ 1.0 x)

   1. Initial program 13.1

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
   2. Using strategy rm

   3. Applied div-inv 13.1

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

   4. Applied times-frac 10.1

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 10.1

      \[\leadsto \frac{1}{\color{blue}{1 \cdot y}} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\]

   7. Applied add-cube-cbrt 10.1

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

   8. Applied times-frac 10.1

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

   9. Applied associate-*l* 10.1

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

   10. Simplified 10.1

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\sqrt[3]{1}}{y}}{\left(1 + z \cdot z\right) \cdot x}}\]
   11. Using strategy rm

   12. Applied div-inv 10.1

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

   13. Applied associate-/l* 10.2

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

   14. Simplified 10.2

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\color{blue}{\left(x \cdot \left({z}^{2} + 1\right)\right) \cdot y}}\]
   15. Using strategy rm

   16. Applied distribute-lft-in 10.2

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

   17. Simplified 3.4

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

   18. Simplified 3.4

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

   if -2.3756245543107074e+20 < (/ 1.0 x) < 2.3441269454524362e+59

   1. Initial program 1.9

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
   2. Using strategy rm

   3. Applied div-inv 1.9

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

   4. Applied times-frac 3.6

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 3.6

      \[\leadsto \frac{1}{\color{blue}{1 \cdot y}} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\]

   7. Applied add-cube-cbrt 3.6

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

   8. Applied times-frac 3.6

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

   9. Applied associate-*l* 3.6

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

   10. Simplified 3.6

       \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \color{blue}{\frac{\frac{\sqrt[3]{1}}{y}}{\left(1 + z \cdot z\right) \cdot x}}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 3.6

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

   13. Applied add-cube-cbrt 3.6

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

   14. Applied times-frac 3.6

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

   15. Applied times-frac 1.8

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

   16. Simplified 1.8

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -237562455431070744576 \lor \neg \left(\frac{1}{x} \le 2.344126945452436231818170630862504221596 \cdot 10^{59}\right):\\ \;\;\;\;\frac{\sqrt[3]{1}}{\left(\left(x \cdot z\right) \cdot z + 1 \cdot x\right) \cdot y} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\frac{\sqrt[3]{\sqrt[3]{1}} \cdot \sqrt[3]{\sqrt[3]{1}}}{{z}^{2} + 1} \cdot \frac{\frac{\sqrt[3]{\sqrt[3]{1}}}{y}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))