Average Error: 6.8 → 6.2
Time: 9.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{\frac{1}{x}}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y}
double f(double x, double y, double z) {
        double r326498 = 1.0;
        double r326499 = x;
        double r326500 = r326498 / r326499;
        double r326501 = y;
        double r326502 = z;
        double r326503 = r326502 * r326502;
        double r326504 = r326498 + r326503;
        double r326505 = r326501 * r326504;
        double r326506 = r326500 / r326505;
        return r326506;
}
↓
double f(double x, double y, double z) {
        double r326507 = 1.0;
        double r326508 = z;
        double r326509 = 1.0;
        double r326510 = fma(r326508, r326508, r326509);
        double r326511 = sqrt(r326510);
        double r326512 = r326507 / r326511;
        double r326513 = x;
        double r326514 = r326509 / r326513;
        double r326515 = r326514 / r326511;
        double r326516 = y;
        double r326517 = r326515 / r326516;
        double r326518 = r326512 * r326517;
        return r326518;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.8
Target6.0
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.8

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

  2. Simplified6.5

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

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

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

  5. Applied add-sqr-sqrt6.5

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

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

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

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

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

  8. Applied times-frac6.5

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

  9. Applied times-frac6.5

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

  10. Applied times-frac6.2

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

  11. Simplified6.2

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

  12. Final simplification6.2

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

Reproduce

herbie shell --seed 2020036 +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)))))