Average Error: 6.4 → 5.9
Time: 1.0m
Precision: 64
\[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{1.0}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}}\]
\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{1.0}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}}
double f(double x, double y, double z) {
        double r16238099 = 1.0;
        double r16238100 = x;
        double r16238101 = r16238099 / r16238100;
        double r16238102 = y;
        double r16238103 = z;
        double r16238104 = r16238103 * r16238103;
        double r16238105 = r16238099 + r16238104;
        double r16238106 = r16238102 * r16238105;
        double r16238107 = r16238101 / r16238106;
        return r16238107;
}
↓
double f(double x, double y, double z) {
        double r16238108 = 1.0;
        double r16238109 = x;
        double r16238110 = r16238108 / r16238109;
        double r16238111 = y;
        double r16238112 = cbrt(r16238111);
        double r16238113 = r16238110 / r16238112;
        double r16238114 = 1.0;
        double r16238115 = z;
        double r16238116 = r16238115 * r16238115;
        double r16238117 = r16238114 + r16238116;
        double r16238118 = sqrt(r16238117);
        double r16238119 = r16238113 / r16238118;
        double r16238120 = r16238112 * r16238112;
        double r16238121 = r16238114 / r16238120;
        double r16238122 = r16238121 / r16238118;
        double r16238123 = r16238119 * r16238122;
        return r16238123;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target5.7
Herbie5.9
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1.0 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1.0 + z \cdot z\right) \lt 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{\left(1.0 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

    \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied associate-/r*6.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1.0}{x}}{y}}{1.0 + z \cdot z}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt6.3

    \[\leadsto \frac{\frac{\frac{1.0}{x}}{y}}{\color{blue}{\sqrt{1.0 + z \cdot z} \cdot \sqrt{1.0 + z \cdot z}}}\]

  6. Applied add-cube-cbrt6.9

    \[\leadsto \frac{\frac{\frac{1.0}{x}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}{\sqrt{1.0 + z \cdot z} \cdot \sqrt{1.0 + z \cdot z}}\]

  7. Applied div-inv6.9

    \[\leadsto \frac{\frac{\color{blue}{1.0 \cdot \frac{1}{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z} \cdot \sqrt{1.0 + z \cdot z}}\]

  8. Applied times-frac6.9

    \[\leadsto \frac{\color{blue}{\frac{1.0}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{1}{x}}{\sqrt[3]{y}}}}{\sqrt{1.0 + z \cdot z} \cdot \sqrt{1.0 + z \cdot z}}\]

  9. Applied times-frac5.9

    \[\leadsto \color{blue}{\frac{\frac{1.0}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}}}\]

  10. Final simplification5.9

    \[\leadsto \frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{1.0}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt{1.0 + z \cdot z}}\]

Reproduce

herbie shell --seed 2019168 
(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)))))