Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.6 → 5.9

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -3.40410512832116911 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \mathbf{elif}\;\frac{1}{x} \le 1.2782103286766604 \cdot 10^{252}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((double) (((double) (1.0 / x)) / ((double) (y * ((double) (1.0 + ((double) (z * z))))))));
}

↓

double code(double x, double y, double z) {
	double VAR;
	if ((((double) (1.0 / x)) <= -3.404105128321169e+99)) {
		VAR = ((double) (((double) (1.0 / y)) * ((double) (((double) (1.0 / x)) / ((double) (1.0 + ((double) (z * z))))))));
	} else {
		double VAR_1;
		if ((((double) (1.0 / x)) <= 1.2782103286766604e+252)) {
			VAR_1 = ((double) (((double) (1.0 / x)) / ((double) (((double) (y * ((double) sqrt(((double) (1.0 + ((double) (z * z)))))))) * ((double) sqrt(((double) (1.0 + ((double) (z * z))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (1.0 / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) / y)) * ((double) (((double) (1.0 / ((double) cbrt(x)))) / ((double) (1.0 + ((double) (z * z))))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.6
Target	5.9
Herbie	5.9

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -inf.0:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 x) < -3.40410512832116911e99

   1. Initial program 16.0

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

   3. Applied *-un-lft-identity 16.0

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

   4. Applied *-un-lft-identity 16.0

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

   5. Applied times-frac 16.0

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

   6. Applied times-frac 12.1

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

   7. Simplified 12.1

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

   if -3.40410512832116911e99 < (/ 1.0 x) < 1.2782103286766604e252

   1. Initial program 4.2

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

   3. Applied add-sqr-sqrt 4.2

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

   4. Applied associate-*r* 4.2

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

   if 1.2782103286766604e252 < (/ 1.0 x)

   1. Initial program 21.2

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

   3. Applied add-cube-cbrt 21.9

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

   4. Applied *-un-lft-identity 21.9

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

   5. Applied times-frac 21.9

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

   6. Applied times-frac 19.5

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -3.40410512832116911 \cdot 10^{99}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \mathbf{elif}\;\frac{1}{x} \le 1.2782103286766604 \cdot 10^{252}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{y} \cdot \frac{\frac{1}{\sqrt[3]{x}}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(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)))))