Average Error: 6.4 → 5.8
Time: 6.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{1}{x}}}{\left(\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}\right) \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{x}}}}{\frac{y}{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{\frac{1}{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]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\sqrt[3]{x}}}}{\frac{y}{\frac{\sqrt[3]{\frac{1}{x}}}{\sqrt[3]{\mathsf{fma}\left(z, z, 1\right)}}}}
double f(double x, double y, double z) {
        double r374535 = 1.0;
        double r374536 = x;
        double r374537 = r374535 / r374536;
        double r374538 = y;
        double r374539 = z;
        double r374540 = r374539 * r374539;
        double r374541 = r374535 + r374540;
        double r374542 = r374538 * r374541;
        double r374543 = r374537 / r374542;
        return r374543;
}
↓
double f(double x, double y, double z) {
        double r374544 = 1.0;
        double r374545 = x;
        double r374546 = r374544 / r374545;
        double r374547 = cbrt(r374546);
        double r374548 = z;
        double r374549 = fma(r374548, r374548, r374544);
        double r374550 = cbrt(r374549);
        double r374551 = r374550 * r374550;
        double r374552 = cbrt(r374545);
        double r374553 = r374552 * r374552;
        double r374554 = cbrt(r374553);
        double r374555 = r374551 * r374554;
        double r374556 = r374547 / r374555;
        double r374557 = cbrt(r374544);
        double r374558 = cbrt(r374552);
        double r374559 = r374557 / r374558;
        double r374560 = r374556 * r374559;
        double r374561 = y;
        double r374562 = r374547 / r374550;
        double r374563 = r374561 / r374562;
        double r374564 = r374560 / r374563;
        return r374564;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.4
Target5.6
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 6.4

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

  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-cube-cbrt6.5

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

  5. Applied add-cube-cbrt6.9

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

  6. Applied times-frac6.9

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

  7. Applied associate-/l*5.8

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

  9. Applied cbrt-div5.7

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

  10. Applied associate-*r/5.7

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

  11. Applied associate-/l/5.7

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

  13. Applied add-cube-cbrt5.8

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

  14. Applied cbrt-prod5.8

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

  15. Applied associate-*r*5.8

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

  17. Applied times-frac5.8

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

  18. Final simplification5.8

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

Reproduce

herbie shell --seed 2020060 +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)))))