Average Error: 6.3 → 5.1
Time: 3.1s
Precision: binary64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -3.2719025730029142 \cdot 10^{+103} \lor \neg \left(x \leq 1.3704778646666072 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -3.2719025730029142 \cdot 10^{+103} \lor \neg \left(x \leq 1.3704778646666072 \cdot 10^{+49}\right):\\
\;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{y}}{x \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 (<= x -3.2719025730029142e+103) (not (<= x 1.3704778646666072e+49)))
   (/ (/ (/ 1.0 x) y) (+ 1.0 (* z z)))
   (/ (/ 1.0 y) (* x (+ 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 tmp;
	if (((x <= -3.2719025730029142e+103) || !(x <= 1.3704778646666072e+49))) {
		tmp = (((1.0 / x) / y) / ((double) (1.0 + ((double) (z * z)))));
	} else {
		tmp = ((1.0 / y) / ((double) (x * ((double) (1.0 + ((double) (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

Original6.3
Target5.6
Herbie5.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.27190257300291421e103 or 1.37047786466660717e49 < x

    1. Initial program 0.9

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

    3. Applied associate-/r*_binary640.9

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

    if -3.27190257300291421e103 < x < 1.37047786466660717e49

    1. Initial program 9.8

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

    3. Applied clear-num_binary6410.0

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

    4. Simplified9.9

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

    6. Applied associate-/r*_binary649.8

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

    8. Applied *-un-lft-identity_binary649.8

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

    9. Applied times-frac_binary649.7

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

    10. Applied associate-/l*_binary647.9

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

    11. Simplified7.9

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

  4. Final simplification5.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2719025730029142 \cdot 10^{+103} \lor \neg \left(x \leq 1.3704778646666072 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{y}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{x \cdot \left(1 + z \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))))