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

Average Error: 14.9 → 2.8

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 9.9114704058449326 \cdot 10^{-126} \lor \neg \left(z \le 2.7466160741339841 \cdot 10^{68}\right):\\ \;\;\;\;{\left(\frac{\frac{x}{z} \cdot \frac{1}{\frac{z + 1}{y}}}{z}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z \cdot \left(z \cdot \left(z + 1\right)\right)}\right)}^{1}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r339759 = x;
        double r339760 = y;
        double r339761 = r339759 * r339760;
        double r339762 = z;
        double r339763 = r339762 * r339762;
        double r339764 = 1.0;
        double r339765 = r339762 + r339764;
        double r339766 = r339763 * r339765;
        double r339767 = r339761 / r339766;
        return r339767;
}

↓

double f(double x, double y, double z) {
        double r339768 = z;
        double r339769 = 9.911470405844933e-126;
        bool r339770 = r339768 <= r339769;
        double r339771 = 2.746616074133984e+68;
        bool r339772 = r339768 <= r339771;
        double r339773 = !r339772;
        bool r339774 = r339770 || r339773;
        double r339775 = x;
        double r339776 = r339775 / r339768;
        double r339777 = 1.0;
        double r339778 = 1.0;
        double r339779 = r339768 + r339778;
        double r339780 = y;
        double r339781 = r339779 / r339780;
        double r339782 = r339777 / r339781;
        double r339783 = r339776 * r339782;
        double r339784 = r339783 / r339768;
        double r339785 = pow(r339784, r339777);
        double r339786 = r339775 * r339780;
        double r339787 = r339768 * r339779;
        double r339788 = r339768 * r339787;
        double r339789 = r339786 / r339788;
        double r339790 = pow(r339789, r339777);
        double r339791 = r339774 ? r339785 : r339790;
        return r339791;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.9
Target	4.2
Herbie	2.8

\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < 9.911470405844933e-126 or 2.746616074133984e+68 < z

   1. Initial program 17.0

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

   3. Applied times-frac 12.4

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

   5. Applied *-un-lft-identity 12.4

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

   6. Applied times-frac 6.0

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

   7. Applied associate-*l* 2.3

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

   9. Applied pow1 2.3

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

   10. Applied pow1 2.3

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

   11. Applied pow-prod-down 2.3

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

   12. Applied pow1 2.3

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

   13. Applied pow-prod-down 2.3

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

   14. Simplified 2.3

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

   16. Applied clear-num 2.4

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

   if 9.911470405844933e-126 < z < 2.746616074133984e+68

   1. Initial program 4.5

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

   3. Applied times-frac 4.8

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

   5. Applied *-un-lft-identity 4.8

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

   6. Applied times-frac 4.9

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

   7. Applied associate-*l* 4.4

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

   9. Applied pow1 4.4

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

   10. Applied pow1 4.4

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

   11. Applied pow-prod-down 4.4

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

   12. Applied pow1 4.4

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

   13. Applied pow-prod-down 4.4

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

   14. Simplified 4.4

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

   16. Applied frac-times 4.5

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

   17. Applied associate-/l/ 4.5

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 9.9114704058449326 \cdot 10^{-126} \lor \neg \left(z \le 2.7466160741339841 \cdot 10^{68}\right):\\ \;\;\;\;{\left(\frac{\frac{x}{z} \cdot \frac{1}{\frac{z + 1}{y}}}{z}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot y}{z \cdot \left(z \cdot \left(z + 1\right)\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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