Average Error: 6.4 → 6.7
Time: 39.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{y \cdot x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{y \cdot x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r438211 = 1.0;
        double r438212 = x;
        double r438213 = r438211 / r438212;
        double r438214 = y;
        double r438215 = z;
        double r438216 = r438215 * r438215;
        double r438217 = r438211 + r438216;
        double r438218 = r438214 * r438217;
        double r438219 = r438213 / r438218;
        return r438219;
}
↓
double f(double x, double y, double z) {
        double r438220 = 1.0;
        double r438221 = y;
        double r438222 = x;
        double r438223 = r438221 * r438222;
        double r438224 = r438220 / r438223;
        double r438225 = z;
        double r438226 = fma(r438225, r438225, r438220);
        double r438227 = sqrt(r438226);
        double r438228 = r438224 / r438227;
        double r438229 = r438228 / r438227;
        return r438229;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.7
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.6

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

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

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

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

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

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

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

  7. Applied times-frac6.6

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

  8. Applied times-frac6.6

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

  9. Simplified6.6

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

  10. Simplified6.7

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

  12. Applied add-sqr-sqrt6.7

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

  13. Applied associate-/r*6.7

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

  14. Simplified6.7

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

  15. Final simplification6.7

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

Reproduce

herbie shell --seed 2020042 +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))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

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