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

Average Error: 6.7 → 6.3

Time: 20.3s

Precision: 64

Math

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

↓

\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}\]

C

double f(double x, double y, double z) {
        double r282968 = 1.0;
        double r282969 = x;
        double r282970 = r282968 / r282969;
        double r282971 = y;
        double r282972 = z;
        double r282973 = r282972 * r282972;
        double r282974 = r282968 + r282973;
        double r282975 = r282971 * r282974;
        double r282976 = r282970 / r282975;
        return r282976;
}

↓

double f(double x, double y, double z) {
        double r282977 = 1.0;
        double r282978 = x;
        double r282979 = cbrt(r282978);
        double r282980 = r282979 * r282979;
        double r282981 = r282977 / r282980;
        double r282982 = y;
        double r282983 = cbrt(r282982);
        double r282984 = r282983 * r282983;
        double r282985 = r282981 / r282984;
        double r282986 = 1.0;
        double r282987 = r282986 / r282979;
        double r282988 = r282987 / r282983;
        double r282989 = z;
        double r282990 = fma(r282989, r282989, r282986);
        double r282991 = r282988 / r282990;
        double r282992 = r282985 * r282991;
        return r282992;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	6.1
Herbie	6.3

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

Derivation

1. Initial program 6.7

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

2. Simplified 6.8

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

4. Applied *-un-lft-identity 6.8

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

5. Applied add-cube-cbrt 7.4

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

6. Applied add-cube-cbrt 7.6

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

7. Applied *-un-lft-identity 7.6

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

8. Applied times-frac 7.6

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

9. Applied times-frac 7.6

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

10. Applied times-frac 6.3

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

11. Simplified 6.3

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

12. Final simplification 6.3

    \[\leadsto \frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\mathsf{fma}\left(z, z, 1\right)}\]

Reproduce

herbie shell --seed 2019323 +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)))))