Average Error: 6.4 → 6.3
Time: 6.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}} \cdot \frac{\frac{1}{x}}{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}\]
\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{1}{x}}{y \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}
double f(double x, double y, double z) {
        double r331093 = 1.0;
        double r331094 = x;
        double r331095 = r331093 / r331094;
        double r331096 = y;
        double r331097 = z;
        double r331098 = r331097 * r331097;
        double r331099 = r331093 + r331098;
        double r331100 = r331096 * r331099;
        double r331101 = r331095 / r331100;
        return r331101;
}
↓
double f(double x, double y, double z) {
        double r331102 = 1.0;
        double r331103 = z;
        double r331104 = 1.0;
        double r331105 = fma(r331103, r331103, r331104);
        double r331106 = sqrt(r331105);
        double r331107 = r331102 / r331106;
        double r331108 = x;
        double r331109 = r331104 / r331108;
        double r331110 = y;
        double r331111 = r331110 * r331106;
        double r331112 = r331109 / r331111;
        double r331113 = r331107 * r331112;
        return r331113;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.8
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.4

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

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

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

  5. Applied add-sqr-sqrt6.4

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

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

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

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

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

  8. Applied times-frac6.4

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

  9. Applied times-frac6.4

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

  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. Using strategy rm

  13. Applied div-inv6.2

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

  14. Applied associate-/l*6.3

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

  15. Simplified6.3

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

  16. Final simplification6.3

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

Reproduce

herbie shell --seed 2020062 +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)))))