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

Average Error: 15.0 → 2.3

Time: 11.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.782439850078144232179047902810204166481 \cdot 10^{92}:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{y}{z}\right) \cdot \frac{1}{1 + z}\\ \mathbf{elif}\;z \le 1.369198787813574730522271894888499920798 \cdot 10^{63}:\\ \;\;\;\;\frac{\frac{1}{z}}{1 + z} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{1 + z}{\frac{y}{z}}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r260666 = x;
        double r260667 = y;
        double r260668 = r260666 * r260667;
        double r260669 = z;
        double r260670 = r260669 * r260669;
        double r260671 = 1.0;
        double r260672 = r260669 + r260671;
        double r260673 = r260670 * r260672;
        double r260674 = r260668 / r260673;
        return r260674;
}

↓

double f(double x, double y, double z) {
        double r260675 = z;
        double r260676 = -2.782439850078144e+92;
        bool r260677 = r260675 <= r260676;
        double r260678 = x;
        double r260679 = r260678 / r260675;
        double r260680 = y;
        double r260681 = r260680 / r260675;
        double r260682 = r260679 * r260681;
        double r260683 = 1.0;
        double r260684 = 1.0;
        double r260685 = r260684 + r260675;
        double r260686 = r260683 / r260685;
        double r260687 = r260682 * r260686;
        double r260688 = 1.3691987878135747e+63;
        bool r260689 = r260675 <= r260688;
        double r260690 = r260683 / r260675;
        double r260691 = r260690 / r260685;
        double r260692 = cbrt(r260678);
        double r260693 = r260692 * r260692;
        double r260694 = r260692 / r260675;
        double r260695 = r260694 * r260680;
        double r260696 = r260693 * r260695;
        double r260697 = r260691 * r260696;
        double r260698 = r260685 / r260681;
        double r260699 = r260683 / r260698;
        double r260700 = r260679 * r260699;
        double r260701 = r260689 ? r260697 : r260700;
        double r260702 = r260677 ? r260687 : r260701;
        return r260702;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.0
Target	3.9
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if z < -2.782439850078144e+92

   1. Initial program 12.7

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

   3. Applied associate-/r* 8.9

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

   4. Simplified 8.9

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z + 1}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 8.9

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

   7. Applied add-sqr-sqrt 64.0

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

   8. Applied unpow-prod-down 64.0

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

   9. Applied times-frac 64.0

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

   10. Applied times-frac 64.0

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

   11. Simplified 64.0

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

   12. Simplified 1.4

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

   14. Applied div-inv 1.5

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

   15. Applied associate-*r* 0.3

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

   if -2.782439850078144e+92 < z < 1.3691987878135747e+63

   1. Initial program 17.3

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

   3. Applied associate-/r* 17.3

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

   4. Simplified 17.3

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z + 1}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 17.3

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

   7. Applied add-sqr-sqrt 42.8

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

   8. Applied unpow-prod-down 42.8

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

   9. Applied times-frac 36.6

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

   10. Applied times-frac 36.5

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

   11. Simplified 36.4

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

   12. Simplified 4.9

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

   14. Applied *-un-lft-identity 4.9

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

   15. Applied div-inv 4.9

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

   16. Applied times-frac 4.9

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

   17. Applied associate-*r* 4.6

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

   18. Simplified 4.6

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

   20. Applied *-un-lft-identity 4.6

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

   21. Applied add-cube-cbrt 5.4

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

   22. Applied times-frac 5.4

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

   23. Applied associate-*l* 3.5

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

   if 1.3691987878135747e+63 < z

   1. Initial program 12.8

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

   3. Applied associate-/r* 9.3

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

   4. Simplified 9.3

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z + 1}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 9.3

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

   7. Applied add-sqr-sqrt 9.4

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

   8. Applied unpow-prod-down 9.4

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

   9. Applied times-frac 1.0

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

   10. Applied times-frac 1.9

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

   11. Simplified 1.8

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

   12. Simplified 1.8

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

   14. Applied *-un-lft-identity 1.8

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

   15. Applied *-un-lft-identity 1.8

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

   16. Applied times-frac 1.8

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

   17. Applied associate-/l* 2.0

       \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{\frac{1}{1}}{\frac{1 + z}{\frac{y}{z}}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.782439850078144232179047902810204166481 \cdot 10^{92}:\\ \;\;\;\;\left(\frac{x}{z} \cdot \frac{y}{z}\right) \cdot \frac{1}{1 + z}\\ \mathbf{elif}\;z \le 1.369198787813574730522271894888499920798 \cdot 10^{63}:\\ \;\;\;\;\frac{\frac{1}{z}}{1 + z} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{1}{\frac{1 + z}{\frac{y}{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(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))))