Average Error: 6.5 → 6.4
Time: 12.7s
Precision: 64
\[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1.0}{\mathsf{fma}\left(z, z, 1.0\right)}}{x}}{y}\]
\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}
↓
\frac{\frac{\frac{1.0}{\mathsf{fma}\left(z, z, 1.0\right)}}{x}}{y}
double f(double x, double y, double z) {
        double r15636837 = 1.0;
        double r15636838 = x;
        double r15636839 = r15636837 / r15636838;
        double r15636840 = y;
        double r15636841 = z;
        double r15636842 = r15636841 * r15636841;
        double r15636843 = r15636837 + r15636842;
        double r15636844 = r15636840 * r15636843;
        double r15636845 = r15636839 / r15636844;
        return r15636845;
}
↓
double f(double x, double y, double z) {
        double r15636846 = 1.0;
        double r15636847 = z;
        double r15636848 = fma(r15636847, r15636847, r15636846);
        double r15636849 = r15636846 / r15636848;
        double r15636850 = x;
        double r15636851 = r15636849 / r15636850;
        double r15636852 = y;
        double r15636853 = r15636851 / r15636852;
        return r15636853;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.9
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1.0 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1.0 + z \cdot z\right) \lt 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{\left(1.0 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

    \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity6.5

    \[\leadsto \frac{\frac{1.0}{\color{blue}{1 \cdot x}}}{y \cdot \left(1.0 + z \cdot z\right)}\]

  4. Applied *-un-lft-identity6.5

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot 1.0}}{1 \cdot x}}{y \cdot \left(1.0 + z \cdot z\right)}\]

  5. Applied times-frac6.5

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{1.0}{x}}}{y \cdot \left(1.0 + z \cdot z\right)}\]

  6. Applied times-frac6.5

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{y} \cdot \frac{\frac{1.0}{x}}{1.0 + z \cdot z}}\]

  7. Simplified6.5

    \[\leadsto \color{blue}{\frac{1}{y}} \cdot \frac{\frac{1.0}{x}}{1.0 + z \cdot z}\]

  8. Simplified6.6

    \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{1.0}{\mathsf{fma}\left(z, z, 1.0\right) \cdot x}}\]
  9. Using strategy rm

  10. Applied associate-*l/6.6

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{1.0}{\mathsf{fma}\left(z, z, 1.0\right) \cdot x}}{y}}\]

  11. Simplified6.4

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

  12. Final simplification6.4

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

Reproduce

herbie shell --seed 2019163 +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)))))