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

Average Error: 6.3 → 0.8

Time: 7.3s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -1.0226122617850201 \cdot 10^{+88}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{\mathsf{hypot}\left(1, z\right)}\\ \frac{\frac{1}{t_0}}{y} \cdot \frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{t_0} \end{array}\\ \mathbf{elif}\;\frac{1}{x} \leq 1.0605848652689036 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot y}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt{\frac{1}{y}}\\ t_1 \cdot \frac{\frac{t_1}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \mathsf{hypot}\left(1, z\right)} \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ 1.0 x) -1.0226122617850201e+88)
   (let* ((t_0 (sqrt (hypot 1.0 z))))
     (* (/ (/ 1.0 t_0) y) (/ (/ (/ 1.0 x) (hypot 1.0 z)) t_0)))
   (if (<= (/ 1.0 x) 1.0605848652689036e+130)
     (/ (/ (/ 1.0 (* x y)) (hypot 1.0 z)) (hypot 1.0 z))
     (let* ((t_1 (sqrt (/ 1.0 y))))
       (* t_1 (/ (/ t_1 (hypot 1.0 z)) (* x (hypot 1.0 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 ((1.0 / x) <= -1.0226122617850201e+88) {
		double t_0_1 = sqrt(hypot(1.0, z));
		tmp = ((1.0 / t_0_1) / y) * (((1.0 / x) / hypot(1.0, z)) / t_0_1);
	} else if ((1.0 / x) <= 1.0605848652689036e+130) {
		tmp = ((1.0 / (x * y)) / hypot(1.0, z)) / hypot(1.0, z);
	} else {
		double t_1 = sqrt(1.0 / y);
		tmp = t_1 * ((t_1 / hypot(1.0, z)) / (x * hypot(1.0, z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	4.9
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 1 x) < -1.0226122617850201e88

   1. Initial program 17.3

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

   2. Simplified 17.3

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

   3. Applied *-un-lft-identity_binary64 17.3

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

   4. Applied times-frac_binary64 13.8

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

   5. Applied add-sqr-sqrt_binary64 13.8

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

   6. Applied *-un-lft-identity_binary64 13.8

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

   7. Applied add-cube-cbrt_binary64 13.8

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

   8. Applied times-frac_binary64 13.8

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

   9. Applied times-frac_binary64 13.8

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

   10. Simplified 13.8

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

   11. Simplified 3.6

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

   12. Applied associate-*l/_binary64 3.6

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

   13. Applied frac-times_binary64 2.5

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

   14. Simplified 2.5

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

   15. Applied add-sqr-sqrt_binary64 2.6

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

   16. Applied *-un-lft-identity_binary64 2.6

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

   17. Applied add-cube-cbrt_binary64 2.6

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

   18. Applied times-frac_binary64 2.6

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

   19. Applied times-frac_binary64 2.6

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

   20. Applied times-frac_binary64 1.1

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

   21. Simplified 1.1

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

   22. Simplified 1.2

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

   if -1.0226122617850201e88 < (/.f64 1 x) < 1.06058486526890363e130

   1. Initial program 2.6

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

   2. Simplified 2.6

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

   3. Applied *-un-lft-identity_binary64 2.6

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

   4. Applied times-frac_binary64 4.1

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

   5. Applied add-sqr-sqrt_binary64 4.1

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

   6. Applied *-un-lft-identity_binary64 4.1

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

   7. Applied add-cube-cbrt_binary64 4.1

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

   8. Applied times-frac_binary64 4.1

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

   9. Applied times-frac_binary64 4.1

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

   10. Simplified 4.1

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

   11. Simplified 3.8

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

   12. Applied associate-*l/_binary64 3.8

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

   13. Applied associate-*r/_binary64 1.1

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

   14. Simplified 0.7

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

   if 1.06058486526890363e130 < (/.f64 1 x)

   1. Initial program 13.7

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

   2. Simplified 13.7

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

   3. Applied *-un-lft-identity_binary64 13.7

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

   4. Applied times-frac_binary64 8.9

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

   5. Applied add-sqr-sqrt_binary64 8.9

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

   6. Applied *-un-lft-identity_binary64 8.9

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

   7. Applied add-cube-cbrt_binary64 8.9

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

   8. Applied times-frac_binary64 8.9

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

   9. Applied times-frac_binary64 8.9

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

   10. Simplified 8.9

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

   11. Simplified 1.3

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

   12. Applied add-sqr-sqrt_binary64 1.5

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

   13. Applied associate-*l*_binary64 1.4

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

   14. Simplified 0.6

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \leq -1.0226122617850201 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{hypot}\left(1, z\right)}}}{y} \cdot \frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{\sqrt{\mathsf{hypot}\left(1, z\right)}}\\ \mathbf{elif}\;\frac{1}{x} \leq 1.0605848652689036 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{\frac{1}{x \cdot y}}{\mathsf{hypot}\left(1, z\right)}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{y}} \cdot \frac{\frac{\sqrt{\frac{1}{y}}}{\mathsf{hypot}\left(1, z\right)}}{x \cdot \mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022093 
(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)))))