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

Average Error: 6.3 → 1.6

Time: 13.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.75057667448614019865650029216263252586 \cdot 10^{57}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z}}{y \cdot z}\\ \mathbf{elif}\;z \le 350365031.3388729095458984375:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z}}{y \cdot z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r10960707 = 1.0;
        double r10960708 = x;
        double r10960709 = r10960707 / r10960708;
        double r10960710 = y;
        double r10960711 = z;
        double r10960712 = r10960711 * r10960711;
        double r10960713 = r10960707 + r10960712;
        double r10960714 = r10960710 * r10960713;
        double r10960715 = r10960709 / r10960714;
        return r10960715;
}

↓

double f(double x, double y, double z) {
        double r10960716 = z;
        double r10960717 = -1.7505766744861402e+57;
        bool r10960718 = r10960716 <= r10960717;
        double r10960719 = 1.0;
        double r10960720 = x;
        double r10960721 = r10960719 / r10960720;
        double r10960722 = r10960721 / r10960716;
        double r10960723 = y;
        double r10960724 = r10960723 * r10960716;
        double r10960725 = r10960722 / r10960724;
        double r10960726 = 350365031.3388729;
        bool r10960727 = r10960716 <= r10960726;
        double r10960728 = fma(r10960716, r10960716, r10960719);
        double r10960729 = r10960728 * r10960723;
        double r10960730 = r10960721 / r10960729;
        double r10960731 = r10960727 ? r10960730 : r10960725;
        double r10960732 = r10960718 ? r10960725 : r10960731;
        return r10960732;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	5.5
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if z < -1.7505766744861402e+57 or 350365031.3388729 < z

   1. Initial program 12.8

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

   2. Taylor expanded around inf 13.0

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

   3. Simplified 2.9

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

   if -1.7505766744861402e+57 < z < 350365031.3388729

   1. Initial program 0.4

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

   2. Taylor expanded around inf 0.4

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

   3. Simplified 0.4

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.75057667448614019865650029216263252586 \cdot 10^{57}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z}}{y \cdot z}\\ \mathbf{elif}\;z \le 350365031.3388729095458984375:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{z}}{y \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))