Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 14.9 → 2.9

Time: 10.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -3.676346882862387632104572682569721878778 \cdot 10^{266} \lor \left(x \cdot y \le -6.86607803514621678723744147253442238451 \cdot 10^{-196} \lor \left(x \cdot y \le -2.030967872118398397344276679320418172136 \cdot 10^{-277} \lor x \cdot y \le 4.760174277986658079177987348647452456326 \cdot 10^{-319}\right)\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z + 1}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{z + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r263502 = x;
        double r263503 = y;
        double r263504 = r263502 * r263503;
        double r263505 = z;
        double r263506 = r263505 * r263505;
        double r263507 = 1.0;
        double r263508 = r263505 + r263507;
        double r263509 = r263506 * r263508;
        double r263510 = r263504 / r263509;
        return r263510;
}

↓

double f(double x, double y, double z) {
        double r263511 = x;
        double r263512 = y;
        double r263513 = r263511 * r263512;
        double r263514 = -3.6763468828623876e+266;
        bool r263515 = r263513 <= r263514;
        double r263516 = -6.866078035146217e-196;
        bool r263517 = r263513 <= r263516;
        double r263518 = -2.0309678721183984e-277;
        bool r263519 = r263513 <= r263518;
        double r263520 = 4.7601742779867e-319;
        bool r263521 = r263513 <= r263520;
        bool r263522 = r263519 || r263521;
        bool r263523 = r263517 || r263522;
        bool r263524 = r263515 || r263523;
        double r263525 = z;
        double r263526 = 1.0;
        double r263527 = r263525 + r263526;
        double r263528 = r263512 / r263527;
        double r263529 = r263528 / r263525;
        double r263530 = r263511 * r263529;
        double r263531 = r263530 / r263525;
        double r263532 = cbrt(r263525);
        double r263533 = r263511 / r263532;
        double r263534 = r263533 * r263528;
        double r263535 = r263532 * r263532;
        double r263536 = r263534 / r263535;
        double r263537 = r263536 / r263525;
        double r263538 = r263524 ? r263531 : r263537;
        return r263538;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.9
Target	4.2
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 5 regimes

2. if (* x y) < -3.6763468828623876e+266

   1. Initial program 53.0

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

   3. Applied times-frac 17.6

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

   5. Applied *-un-lft-identity 17.6

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

   6. Applied times-frac 2.1

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

   7. Applied associate-*l* 5.7

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

   9. Applied add-cube-cbrt 6.4

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

   if -3.6763468828623876e+266 < (* x y) < -6.866078035146217e-196

   1. Initial program 6.2

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

   3. Applied associate-/r* 3.5

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

   if -6.866078035146217e-196 < (* x y) < -2.0309678721183984e-277

   1. Initial program 13.3

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

   3. Applied times-frac 14.0

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

   5. Applied *-un-lft-identity 14.0

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

   6. Applied times-frac 8.3

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

   7. Applied associate-*l* 0.6

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

   9. Applied div-inv 0.7

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

   10. Applied associate-*l* 0.7

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

   11. Simplified 0.6

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

   13. Applied *-un-lft-identity 0.6

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

   14. Applied associate-*l* 0.6

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

   15. Simplified 0.6

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

   if -2.0309678721183984e-277 < (* x y) < 4.7601742779867e-319

   1. Initial program 25.5

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

   3. Applied times-frac 15.8

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

   5. Applied associate-/r* 5.3

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

   if 4.7601742779867e-319 < (* x y)

   1. Initial program 12.9

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

   3. Applied times-frac 10.6

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

   5. Applied *-un-lft-identity 10.6

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

   6. Applied times-frac 6.6

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

   7. Applied associate-*l* 2.2

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

   9. Applied add-cube-cbrt 2.7

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

   10. Applied *-un-lft-identity 2.7

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

   11. Applied times-frac 2.7

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

   12. Applied associate-*l* 1.9

       \[\leadsto \frac{1}{z} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \left(\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{z + 1}\right)\right)}\]
3. Recombined 5 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -3.676346882862387632104572682569721878778 \cdot 10^{266} \lor \left(x \cdot y \le -6.86607803514621678723744147253442238451 \cdot 10^{-196} \lor \left(x \cdot y \le -2.030967872118398397344276679320418172136 \cdot 10^{-277} \lor x \cdot y \le 4.760174277986658079177987348647452456326 \cdot 10^{-319}\right)\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z + 1}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{\sqrt[3]{z}} \cdot \frac{y}{z + 1}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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