Average Error: 6.6 → 2.7
Time: 2.8s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 1.501432873824825 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(1 \cdot x\right) + y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 1.501432873824825 \cdot 10^{+258}\right):\\
\;\;\;\;\frac{1}{y \cdot \left(1 \cdot x\right) + y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(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 (<= (* y (+ 1.0 (* z z))) (- INFINITY))
         (not (<= (* y (+ 1.0 (* z z))) 1.501432873824825e+258)))
   (/ 1.0 (+ (* y (* 1.0 x)) (* y (* z (* z x)))))
   (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))))
double code(double x, double y, double z) {
	return ((1.0 / x) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))));
}
↓
double code(double x, double y, double z) {
	double VAR;
	if (((((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= ((double) -(((double) INFINITY)))) || !(((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= 1.501432873824825e+258))) {
		VAR = (1.0 / ((double) (((double) (y * ((double) (1.0 * x)))) + ((double) (y * ((double) (z * ((double) (z * x)))))))));
	} else {
		VAR = ((1.0 / x) / ((double) (y * ((double) (1.0 + ((double) (z * z)))))));
	}
	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

Original6.6
Target5.9
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 (* y (+ 1.0 (* z z))) < -inf.0 or 1.50143287382482497e258 < (* y (+ 1.0 (* z z)))

    1. Initial program 17.0

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

    2. Simplified17.0

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

    4. Applied distribute-lft-in17.0

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

    5. Applied distribute-lft-in17.0

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

    6. Simplified17.0

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

    7. Simplified6.5

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

    if -inf.0 < (* y (+ 1.0 (* z z))) < 1.50143287382482497e258

    1. Initial program 0.3

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 1.501432873824825 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(1 \cdot x\right) + y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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)))))