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

Average Error: 6.5 → 5.5

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -2.8963162947031292 \cdot 10^{-71} \lor \neg \left(\frac{1}{x} \le 1.4134354570333115 \cdot 10^{231}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r370800 = 1.0;
        double r370801 = x;
        double r370802 = r370800 / r370801;
        double r370803 = y;
        double r370804 = z;
        double r370805 = r370804 * r370804;
        double r370806 = r370800 + r370805;
        double r370807 = r370803 * r370806;
        double r370808 = r370802 / r370807;
        return r370808;
}

↓

double f(double x, double y, double z) {
        double r370809 = 1.0;
        double r370810 = x;
        double r370811 = r370809 / r370810;
        double r370812 = -2.896316294703129e-71;
        bool r370813 = r370811 <= r370812;
        double r370814 = 1.4134354570333115e+231;
        bool r370815 = r370811 <= r370814;
        double r370816 = !r370815;
        bool r370817 = r370813 || r370816;
        double r370818 = z;
        double r370819 = fma(r370818, r370818, r370809);
        double r370820 = r370811 / r370819;
        double r370821 = y;
        double r370822 = r370820 / r370821;
        double r370823 = r370809 / r370821;
        double r370824 = r370823 / r370810;
        double r370825 = r370818 * r370818;
        double r370826 = r370809 + r370825;
        double r370827 = r370824 / r370826;
        double r370828 = r370817 ? r370822 : r370827;
        return r370828;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	5.8
Herbie	5.5

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (/ 1.0 x) < -2.896316294703129e-71 or 1.4134354570333115e+231 < (/ 1.0 x)

   1. Initial program 11.4

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

   2. Simplified 9.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}}\]

   if -2.896316294703129e-71 < (/ 1.0 x) < 1.4134354570333115e+231

   1. Initial program 4.0

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

   3. Applied associate-/r* 3.7

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

   4. Simplified 3.7

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -2.8963162947031292 \cdot 10^{-71} \lor \neg \left(\frac{1}{x} \le 1.4134354570333115 \cdot 10^{231}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))))