Average Error: 6.7 → 2.7
Time: 8.2s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -7.533396507028953 \cdot 10^{-29} \lor \neg \left(\frac{1}{x} \leq -6.450279130107457 \cdot 10^{-261}\right):\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \leq -7.533396507028953 \cdot 10^{-29} \lor \neg \left(\frac{1}{x} \leq -6.450279130107457 \cdot 10^{-261}\right):\\
\;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\

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

\end{array}
(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (or (<= (/ 1.0 x) -7.533396507028953e-29)
         (not (<= (/ 1.0 x) -6.450279130107457e-261)))
   (/ (/ 1.0 y) (+ x (* z (* x z))))
   (/ (/ 1.0 x) (* (sqrt (+ 1.0 (* z z))) (* y (sqrt (+ 1.0 (* z z))))))))
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) <= -7.533396507028953e-29) || !((1.0 / x) <= -6.450279130107457e-261)) {
		tmp = (1.0 / y) / (x + (z * (x * z)));
	} else {
		tmp = (1.0 / x) / (sqrt(1.0 + (z * z)) * (y * sqrt(1.0 + (z * z))));
	}
	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

Original6.7
Target5.2
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 2 regimes

  2. if (/.f64 1 x) < -7.5333965070289534e-29 or -6.45027913010745728e-261 < (/.f64 1 x)

    1. Initial program 9.4

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

    3. Applied *-un-lft-identity_binary64_109909.4

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

    4. Applied add-sqr-sqrt_binary64_110129.4

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

    5. Applied times-frac_binary64_109969.4

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

    6. Applied times-frac_binary64_109967.6

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

    7. Simplified7.6

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

    8. Simplified7.7

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

    10. Applied associate-*l*_binary64_109313.4

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

    11. Simplified3.4

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

    13. Applied un-div-inv_binary64_109883.3

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

    if -7.5333965070289534e-29 < (/.f64 1 x) < -6.45027913010745728e-261

    1. Initial program 1.5

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

    3. Applied add-sqr-sqrt_binary64_110121.6

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

    4. Applied associate-*r*_binary64_109301.6

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -7.533396507028953 \cdot 10^{-29} \lor \neg \left(\frac{1}{x} \leq -6.450279130107457 \cdot 10^{-261}\right):\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z} \cdot \left(y \cdot \sqrt{1 + z \cdot z}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021096 
(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)))))