Average Error: 5.9 → 5.5
Time: 1.3m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot 1\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\frac{1}{y}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot 1
double f(double x, double y, double z) {
        double r14779316 = 1.0;
        double r14779317 = x;
        double r14779318 = r14779316 / r14779317;
        double r14779319 = y;
        double r14779320 = z;
        double r14779321 = r14779320 * r14779320;
        double r14779322 = r14779316 + r14779321;
        double r14779323 = r14779319 * r14779322;
        double r14779324 = r14779318 / r14779323;
        return r14779324;
}
↓
double f(double x, double y, double z) {
        double r14779325 = 1.0;
        double r14779326 = y;
        double r14779327 = r14779325 / r14779326;
        double r14779328 = z;
        double r14779329 = 1.0;
        double r14779330 = fma(r14779328, r14779328, r14779329);
        double r14779331 = sqrt(r14779330);
        double r14779332 = r14779327 / r14779331;
        double r14779333 = x;
        double r14779334 = r14779332 / r14779333;
        double r14779335 = r14779334 / r14779331;
        double r14779336 = r14779335 * r14779329;
        return r14779336;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original5.9
Target5.3
Herbie5.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 5.9

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

  2. Simplified6.0

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

  4. Applied add-sqr-sqrt6.0

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

  5. Applied div-inv6.1

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

  6. Applied times-frac5.6

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

  8. Applied *-un-lft-identity5.6

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

  9. Applied div-inv5.6

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

  10. Applied times-frac5.6

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

  11. Applied associate-*l*5.6

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

  12. Simplified5.5

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

  13. Final simplification5.5

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

Reproduce

herbie shell --seed 2019169 +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)))))