Average Error: 6.5 → 5.8
Time: 14.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}{\frac{y}{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}
double f(double x, double y, double z) {
        double r1433 = 1.0;
        double r1434 = x;
        double r1435 = r1433 / r1434;
        double r1436 = y;
        double r1437 = z;
        double r1438 = r1437 * r1437;
        double r1439 = r1433 + r1438;
        double r1440 = r1436 * r1439;
        double r1441 = r1435 / r1440;
        return r1441;
}
↓
double f(double x, double y, double z) {
        double r1442 = 1.0;
        double r1443 = cbrt(r1442);
        double r1444 = r1443 * r1443;
        double r1445 = x;
        double r1446 = cbrt(r1445);
        double r1447 = r1446 * r1446;
        double r1448 = r1444 / r1447;
        double r1449 = z;
        double r1450 = fma(r1449, r1449, r1442);
        double r1451 = cbrt(r1450);
        double r1452 = r1451 * r1451;
        double r1453 = r1448 / r1452;
        double r1454 = y;
        double r1455 = r1443 / r1446;
        double r1456 = r1455 / r1451;
        double r1457 = r1454 / r1456;
        double r1458 = r1453 / r1457;
        return r1458;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.5
Target5.7
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  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 add-cube-cbrt6.6

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

  5. Applied add-cube-cbrt7.1

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

  6. Applied add-cube-cbrt7.1

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

  7. Applied times-frac7.1

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

  8. Applied times-frac7.1

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

  9. Applied associate-/l*5.8

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

  10. Final simplification5.8

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

Reproduce

herbie shell --seed 2020025 +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)))))