Average Error: 6.6 → 6.5
Time: 4.7m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(z, z, 1\right) \cdot \frac{1}{\frac{\sqrt{1}}{x}}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\mathsf{fma}\left(z, z, 1\right) \cdot \frac{1}{\frac{\sqrt{1}}{x}}}}{y}
double f(double x, double y, double z) {
        double r977447 = 1.0;
        double r977448 = x;
        double r977449 = r977447 / r977448;
        double r977450 = y;
        double r977451 = z;
        double r977452 = r977451 * r977451;
        double r977453 = r977447 + r977452;
        double r977454 = r977450 * r977453;
        double r977455 = r977449 / r977454;
        return r977455;
}
↓
double f(double x, double y, double z) {
        double r977456 = 1.0;
        double r977457 = sqrt(r977456);
        double r977458 = z;
        double r977459 = fma(r977458, r977458, r977456);
        double r977460 = 1.0;
        double r977461 = x;
        double r977462 = r977457 / r977461;
        double r977463 = r977460 / r977462;
        double r977464 = r977459 * r977463;
        double r977465 = r977457 / r977464;
        double r977466 = y;
        double r977467 = r977465 / r977466;
        return r977467;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.6
Target5.9
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

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 add-sqr-sqrt6.9

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

  7. Applied times-frac6.9

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

  8. Applied times-frac6.9

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

  9. Applied associate-/l*7.3

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

  11. Applied associate-/r/6.8

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

  12. Applied associate-/r*6.5

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

  14. Applied div-inv6.5

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

  15. Final simplification6.5

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

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)))))