Average Error: 6.6 → 6.6
Time: 2.6m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{y} \cdot \frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{y} \cdot \frac{1}{\frac{\mathsf{fma}\left(z, z, 1\right)}{\frac{1}{x}}}
double f(double x, double y, double z) {
        double r555359 = 1.0;
        double r555360 = x;
        double r555361 = r555359 / r555360;
        double r555362 = y;
        double r555363 = z;
        double r555364 = r555363 * r555363;
        double r555365 = r555359 + r555364;
        double r555366 = r555362 * r555365;
        double r555367 = r555361 / r555366;
        return r555367;
}
↓
double f(double x, double y, double z) {
        double r555368 = 1.0;
        double r555369 = y;
        double r555370 = r555368 / r555369;
        double r555371 = z;
        double r555372 = 1.0;
        double r555373 = fma(r555371, r555371, r555372);
        double r555374 = x;
        double r555375 = r555372 / r555374;
        double r555376 = r555373 / r555375;
        double r555377 = r555368 / r555376;
        double r555378 = r555370 * r555377;
        return r555378;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target5.9
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  2. Simplified6.9

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

  4. Applied *-un-lft-identity6.9

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

  5. Applied *-un-lft-identity6.9

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

  6. Applied *-un-lft-identity6.9

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

  7. Applied times-frac6.9

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

  8. Applied times-frac6.9

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

  9. Applied associate-/l*7.3

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

  11. Applied associate-/r/6.8

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

  12. Applied add-sqr-sqrt6.8

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

  13. Applied add-sqr-sqrt6.8

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

  14. Applied times-frac6.8

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

  15. Applied times-frac6.6

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

  16. Final simplification6.6

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

Reproduce

herbie shell --seed 2019303 +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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))