Average Error: 6.4 → 3.4
Time: 12.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\
\;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\

\end{array}
double f(double x, double y, double z) {
        double r260098 = 1.0;
        double r260099 = x;
        double r260100 = r260098 / r260099;
        double r260101 = y;
        double r260102 = z;
        double r260103 = r260102 * r260102;
        double r260104 = r260098 + r260103;
        double r260105 = r260101 * r260104;
        double r260106 = r260100 / r260105;
        return r260106;
}
↓
double f(double x, double y, double z) {
        double r260107 = z;
        double r260108 = r260107 * r260107;
        double r260109 = 4.652915945381897e+233;
        bool r260110 = r260108 <= r260109;
        double r260111 = 1.0;
        double r260112 = 1.0;
        double r260113 = y;
        double r260114 = r260112 / r260113;
        double r260115 = r260111 * r260114;
        double r260116 = r260111 + r260108;
        double r260117 = x;
        double r260118 = r260116 * r260117;
        double r260119 = r260115 / r260118;
        double r260120 = r260111 / r260117;
        double r260121 = r260113 * r260107;
        double r260122 = r260107 * r260121;
        double r260123 = r260120 / r260122;
        double r260124 = 4.0;
        double r260125 = pow(r260107, r260124);
        double r260126 = r260113 * r260125;
        double r260127 = r260120 / r260126;
        double r260128 = r260123 - r260127;
        double r260129 = r260110 ? r260119 : r260128;
        return r260129;
}

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
Target5.8
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 4.652915945381897e+233

    1. Initial program 1.6

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied clear-num2.0

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

    4. Simplified1.9

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{1} \cdot \left(\left(1 + z \cdot z\right) \cdot x\right)}}\]
    5. Using strategy rm

    6. Applied associate-/r*1.6

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

    7. Simplified1.6

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

    if 4.652915945381897e+233 < (* z z)

    1. Initial program 16.3

      \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
    2. Using strategy rm

    3. Applied clear-num16.4

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

    4. Simplified16.6

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

    5. Taylor expanded around inf 16.3

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot \left({z}^{2} \cdot y\right)} - 1 \cdot \frac{1}{x \cdot \left({z}^{4} \cdot y\right)}}\]

    6. Simplified7.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))))