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

Average Error: 6.9 → 6.4

Time: 59.0s

Precision: 64

Math

\[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]

↓

\[\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}}\right)\]

C

double f(double x, double y, double z) {
        double r17288762 = 1.0;
        double r17288763 = x;
        double r17288764 = r17288762 / r17288763;
        double r17288765 = y;
        double r17288766 = z;
        double r17288767 = r17288766 * r17288766;
        double r17288768 = r17288762 + r17288767;
        double r17288769 = r17288765 * r17288768;
        double r17288770 = r17288764 / r17288769;
        return r17288770;
}

↓

double f(double x, double y, double z) {
        double r17288771 = 1.0;
        double r17288772 = cbrt(r17288771);
        double r17288773 = x;
        double r17288774 = cbrt(r17288773);
        double r17288775 = r17288772 / r17288774;
        double r17288776 = z;
        double r17288777 = fma(r17288776, r17288776, r17288771);
        double r17288778 = cbrt(r17288777);
        double r17288779 = r17288775 / r17288778;
        double r17288780 = y;
        double r17288781 = cbrt(r17288780);
        double r17288782 = r17288779 / r17288781;
        double r17288783 = r17288782 * r17288782;
        double r17288784 = r17288782 * r17288783;
        return r17288784;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.9
Target	6.2
Herbie	6.4

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1.0 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1.0 + z \cdot z\right) \lt 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{\left(1.0 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Initial program 6.9

   \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]

2. Simplified 6.8

   \[\leadsto \color{blue}{\frac{\frac{\frac{1.0}{x}}{y}}{\mathsf{fma}\left(z, z, 1.0\right)}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 6.9

   \[\leadsto \frac{\frac{\frac{1.0}{x}}{y}}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}\]

5. Applied add-cube-cbrt 7.4

   \[\leadsto \frac{\frac{\frac{1.0}{x}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

6. Applied add-cube-cbrt 7.6

   \[\leadsto \frac{\frac{\frac{1.0}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

7. Applied add-cube-cbrt 7.6

   \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}\right) \cdot \sqrt[3]{1.0}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

8. Applied times-frac 7.6

   \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

9. Applied times-frac 7.6

   \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

10. Applied times-frac 6.5

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{1.0} \cdot \sqrt[3]{1.0}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}\]

11. Simplified 6.4

    \[\leadsto \color{blue}{\left(\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}}\right)} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}\]

12. Simplified 6.4

    \[\leadsto \left(\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}}\right) \cdot \color{blue}{\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}}}\]

13. Final simplification 6.4

    \[\leadsto \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \left(\frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt[3]{1.0}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1.0\right)}}}{\sqrt[3]{y}}\right)\]

Reproduce

herbie shell --seed 2019158 +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)))))