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

Average Error: 6.5 → 5.2

Time: 3.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3941269482623205 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}}\\ \mathbf{elif}\;y \leq 2231.9134022336957:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.3941269482623205e-62)
   (*
    (/ 1.0 (sqrt (+ 1.0 (* z z))))
    (/ (/ (/ 1.0 x) y) (sqrt (+ 1.0 (* z z)))))
   (if (<= y 2231.9134022336957)
     (/ 1.0 (* x (* y (+ 1.0 (* z z)))))
     (* (/ 1.0 y) (/ (/ 1.0 x) (+ 1.0 (* z z)))))))

C

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 (y <= -5.3941269482623205e-62) {
		tmp = (1.0 / sqrt(1.0 + (z * z))) * (((1.0 / x) / y) / sqrt(1.0 + (z * z)));
	} else if (y <= 2231.9134022336957) {
		tmp = 1.0 / (x * (y * (1.0 + (z * z))));
	} else {
		tmp = (1.0 / y) * ((1.0 / x) / (1.0 + (z * z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.5
Target	5.7
Herbie	5.2

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

Derivation

1. Split input into 3 regimes

2. if y < -5.39412694826232054e-62

   1. Initial program 3.9

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

   3. Applied associate-/r*_binary64 2.0

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

   5. Applied add-sqr-sqrt_binary64 2.1

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

   6. Applied *-un-lft-identity_binary64 2.1

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

   7. Applied *-un-lft-identity_binary64 2.1

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

   8. Applied add-sqr-sqrt_binary64 2.1

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

   9. Applied times-frac_binary64 2.1

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

   10. Applied times-frac_binary64 2.1

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

   11. Applied times-frac_binary64 2.1

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

   12. Simplified 2.1

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

   13. Simplified 2.1

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

   if -5.39412694826232054e-62 < y < 2231.9134022336957

   1. Initial program 9.7

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

   3. Applied clear-num_binary64 9.8

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

   4. Simplified 9.8

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

   if 2231.9134022336957 < y

   1. Initial program 4.4

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

   3. Applied *-un-lft-identity_binary64 4.4

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

   4. Applied add-sqr-sqrt_binary64 4.4

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

   5. Applied times-frac_binary64 4.4

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

   6. Applied times-frac_binary64 1.7

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

   7. Simplified 1.7

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

   8. Simplified 1.7

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

4. Final simplification 5.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.3941269482623205 \cdot 10^{-62}:\\ \;\;\;\;\frac{1}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{x}}{y}}{\sqrt{1 + z \cdot z}}\\ \mathbf{elif}\;y \leq 2231.9134022336957:\\ \;\;\;\;\frac{1}{x \cdot \left(y \cdot \left(1 + z \cdot z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{\frac{1}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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)))))