Average Error: 6.5 → 2.7
Time: 7.1s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -388312472441559.06:\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{elif}\;\frac{1}{x} \leq 5.552385065355027 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)\right)}^{-1}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \leq -388312472441559.06:\\
\;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\

\mathbf{elif}\;\frac{1}{x} \leq 5.552385065355027 \cdot 10^{+164}:\\
\;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= (/ 1.0 x) -388312472441559.06)
   (/ (/ 1.0 y) (+ x (* z (* x z))))
   (if (<= (/ 1.0 x) 5.552385065355027e+164)
     (/ (/ 1.0 x) (* (sqrt (+ 1.0 (* z z))) (* y (sqrt (+ 1.0 (* z z))))))
     (pow (* y (+ x (* z (* x z)))) -1.0))))
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 tmp;
	if ((1.0 / x) <= -388312472441559.06) {
		tmp = (1.0 / y) / (x + (z * (x * z)));
	} else if ((1.0 / x) <= 5.552385065355027e+164) {
		tmp = (1.0 / x) / (sqrt(1.0 + (z * z)) * (y * sqrt(1.0 + (z * z))));
	} else {
		tmp = pow((y * (x + (z * (x * z)))), -1.0);
	}
	return tmp;
}

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

Original6.5
Target5.1
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 1 x) < -388312472441559.062

    1. Initial program 13.6

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

    3. Applied *-un-lft-identity_binary64_1133113.6

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

    4. Applied add-sqr-sqrt_binary64_1135313.6

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

    5. Applied times-frac_binary64_1133713.6

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

    6. Applied times-frac_binary64_1133710.9

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

    7. Simplified10.9

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

    8. Simplified10.9

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

    10. Applied associate-*r*_binary64_112714.3

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

    12. Applied un-div-inv_binary64_113294.2

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

    if -388312472441559.062 < (/.f64 1 x) < 5.552385065355027e164

    1. Initial program 2.3

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

    3. Applied add-sqr-sqrt_binary64_113532.3

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

    4. Applied associate-*r*_binary64_112712.3

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

    if 5.552385065355027e164 < (/.f64 1 x)

    1. Initial program 17.2

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

    3. Applied *-un-lft-identity_binary64_1133117.2

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

    4. Applied add-sqr-sqrt_binary64_1135317.2

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

    5. Applied times-frac_binary64_1133717.2

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

    6. Applied times-frac_binary64_1133711.8

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

    7. Simplified11.8

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

    8. Simplified11.8

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

    10. Applied associate-*r*_binary64_112711.6

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

    12. Applied inv-pow_binary64_114161.6

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

    13. Applied inv-pow_binary64_114161.6

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

    14. Applied pow-prod-down_binary64_114021.8

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -388312472441559.06:\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{elif}\;\frac{1}{x} \leq 5.552385065355027 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(y \cdot \left(x + z \cdot \left(x \cdot z\right)\right)\right)}^{-1}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 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)))))