Average Error: 6.7 → 2.5
Time: 3.6s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 1.177368435507795 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\ \end{array}\]
\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 1.177368435507795 \cdot 10^{+304}\right):\\
\;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\

\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))) 1.177368435507795e+304)))
   (/ 1.0 (* y (* z (* z x))))
   (/ (/ (/ 1.0 y) (+ 1.0 (* z z))) x)))
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 tmp;
	if (((((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= ((double) -(((double) INFINITY)))) || !(((double) (y * ((double) (1.0 + ((double) (z * z)))))) <= 1.177368435507795e+304))) {
		tmp = (1.0 / ((double) (y * ((double) (z * ((double) (z * x)))))));
	} else {
		tmp = (((1.0 / y) / ((double) (1.0 + ((double) (z * z))))) / x);
	}
	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
Target6.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* y (+ 1.0 (* z z))) < -inf.0 or 1.1773684355077949e304 < (* y (+ 1.0 (* z z)))

    1. Initial program Error: 19.2 bits

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

    3. Applied div-invError: 19.2 bits

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

    4. Taylor expanded around inf Error: 19.4 bits

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

    5. SimplifiedError: 6.9 bits

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

    if -inf.0 < (* y (+ 1.0 (* z z))) < 1.1773684355077949e304

    1. Initial program Error: 0.3 bits

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

    3. Applied div-invError: 0.3 bits

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

    5. Applied associate-*l/Error: 0.3 bits

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

    6. SimplifiedError: 0.3 bits

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

    8. Applied associate-/r*Error: 0.3 bits

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

  4. Final simplificationError: 2.5 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq -\infty \lor \neg \left(y \cdot \left(1 + z \cdot z\right) \leq 1.177368435507795 \cdot 10^{+304}\right):\\ \;\;\;\;\frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{1 + z \cdot z}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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)))))