Average Error: 6.4 → 5.1
Time: 4.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.152553010188792 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}\\ \mathbf{elif}\;y \le 4.337130840969958 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.152553010188792 \cdot 10^{-160}:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}\\

\mathbf{elif}\;y \le 4.337130840969958 \cdot 10^{-21}:\\
\;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\

\end{array}
double code(double x, double y, double z) {
	return ((1.0 / x) / (y * (1.0 + (z * z))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if ((y <= -1.152553010188792e-160)) {
		VAR = ((1.0 / sqrt(fma(z, z, 1.0))) / (y * (sqrt(fma(z, z, 1.0)) * x)));
	} else {
		double VAR_1;
		if ((y <= 4.337130840969958e-21)) {
			VAR_1 = ((1.0 / x) * ((1.0 / fma(z, z, 1.0)) / y));
		} else {
			VAR_1 = ((1.0 / (fma(z, z, 1.0) * x)) / y);
		}
		VAR = VAR_1;
	}
	return VAR;
}

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

Derivation

  1. Split input into 3 regimes

  2. if y < -1.152553010188792e-160

    1. Initial program 4.5

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

    2. Simplified3.1

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

    4. Applied add-sqr-sqrt3.1

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

    5. Applied div-inv3.1

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

    6. Applied times-frac3.1

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

    7. Applied associate-/l*3.0

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

    8. Simplified3.0

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

    if -1.152553010188792e-160 < y < 4.337130840969958e-21

    1. Initial program 10.7

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

    2. Simplified14.0

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

    4. Applied *-un-lft-identity14.0

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

    5. Applied div-inv14.0

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

    6. Applied times-frac10.8

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

    7. Simplified10.8

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

    if 4.337130840969958e-21 < y

    1. Initial program 4.5

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

    2. Simplified1.8

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

    4. Applied div-inv1.8

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

    5. Applied associate-/l*1.9

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

    6. Simplified1.8

      \[\leadsto \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}}{y}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.152553010188792 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(z, z, 1\right)}}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)}\\ \mathbf{elif}\;y \le 4.337130840969958 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(z, z, 1\right) \cdot x}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))))