Average Error: 6.3 → 1.5
Time: 19.3s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -2.1954875944739658 \cdot 10^{+55} \lor \neg \left(\frac{1}{x} \leq 105588422183978.67\right):\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{\sqrt{1}}{x}}{y + z \cdot \left(y \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{x} \leq -2.1954875944739658 \cdot 10^{+55} \lor \neg \left(\frac{1}{x} \leq 105588422183978.67\right):\\
\;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1} \cdot \frac{\frac{\sqrt{1}}{x}}{y + z \cdot \left(y \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) -2.1954875944739658e+55)
         (not (<= (/ 1.0 x) 105588422183978.67)))
   (/ (/ 1.0 y) (+ x (* z (* x z))))
   (* (sqrt 1.0) (/ (/ (sqrt 1.0) x) (+ y (* z (* y 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) <= -2.1954875944739658e+55) || !((1.0 / x) <= 105588422183978.67)) {
		tmp = (1.0 / y) / (x + (z * (x * z)));
	} else {
		tmp = sqrt(1.0) * ((sqrt(1.0) / x) / (y + (z * (y * 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

Original6.3
Target5.8
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 1 x) < -2.1954875944739658e55 or 105588422183978.672 < (/.f64 1 x)

    1. Initial program 12.9

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

    2. Simplified13.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}}\]
    3. Using strategy rm

    4. Applied associate-/l/_binary64_1639312.9

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

    5. Simplified8.9

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

    6. Taylor expanded around 0 12.9

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

    7. Simplified3.3

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

    9. Applied associate-/r*_binary64_163903.1

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

    if -2.1954875944739658e55 < (/.f64 1 x) < 105588422183978.672

    1. Initial program 1.9

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

    2. Simplified1.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}}\]
    3. Using strategy rm

    4. Applied *-un-lft-identity_binary64_164461.9

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

    5. Applied *-un-lft-identity_binary64_164461.9

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

    6. Applied *-un-lft-identity_binary64_164461.9

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

    7. Applied add-sqr-sqrt_binary64_164681.9

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

    8. Applied times-frac_binary64_164521.9

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

    9. Applied times-frac_binary64_164521.9

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

    10. Applied times-frac_binary64_164521.9

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

    11. Simplified0.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -2.1954875944739658 \cdot 10^{+55} \lor \neg \left(\frac{1}{x} \leq 105588422183978.67\right):\\ \;\;\;\;\frac{\frac{1}{y}}{x + z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot \frac{\frac{\sqrt{1}}{x}}{y + z \cdot \left(y \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))))