Average Error: 6.1 → 6.1
Time: 3.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\frac{\sqrt{1}}{\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\frac{\sqrt{1}}{\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}}
double f(double x, double y, double z) {
        double r264395 = 1.0;
        double r264396 = x;
        double r264397 = r264395 / r264396;
        double r264398 = y;
        double r264399 = z;
        double r264400 = r264399 * r264399;
        double r264401 = r264395 + r264400;
        double r264402 = r264398 * r264401;
        double r264403 = r264397 / r264402;
        return r264403;
}
↓
double f(double x, double y, double z) {
        double r264404 = 1.0;
        double r264405 = sqrt(r264404);
        double r264406 = z;
        double r264407 = fma(r264406, r264406, r264404);
        double r264408 = sqrt(r264407);
        double r264409 = sqrt(r264408);
        double r264410 = r264405 / r264409;
        double r264411 = y;
        double r264412 = x;
        double r264413 = r264408 * r264412;
        double r264414 = r264411 * r264413;
        double r264415 = cbrt(r264408);
        double r264416 = r264415 * r264415;
        double r264417 = r264416 * r264415;
        double r264418 = sqrt(r264417);
        double r264419 = r264405 / r264418;
        double r264420 = r264414 / r264419;
        double r264421 = r264410 / r264420;
        return r264421;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.1
Target5.5
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  2. Simplified6.3

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

  4. Applied add-sqr-sqrt6.3

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

  5. Applied div-inv6.3

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

  6. Applied times-frac6.3

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

  7. Applied associate-/l*6.0

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

  8. Simplified6.1

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

  10. Applied add-sqr-sqrt6.1

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

  11. Applied sqrt-prod6.1

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

  12. Applied add-sqr-sqrt6.1

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

  13. Applied times-frac6.1

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

  14. Applied associate-/l*6.1

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

  16. Applied add-cube-cbrt6.1

    \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\frac{\sqrt{1}}{\sqrt{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}}}\]

  17. Final simplification6.1

    \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}{\frac{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}{\frac{\sqrt{1}}{\sqrt{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}}}}\]

Reproduce

herbie shell --seed 2020047 +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)))))