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

Average Error: 6.7 → 0.9

Time: 9.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{if}\;t_0 \leq -4.380498551296909 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}\\ \mathbf{elif}\;t_0 \leq 6.697308091806901 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(1 + {z}^{2}\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 x) (* y (+ 1.0 (* z z))))))
   (if (<= t_0 -4.380498551296909e-285)
     (/ (/ 1.0 x) (* (* y (hypot 1.0 z)) (sqrt (fma z z 1.0))))
     (if (<= t_0 6.697308091806901e-18)
       (/ (/ (/ (/ 1.0 x) (hypot 1.0 z)) y) (hypot 1.0 z))
       (/ 1.0 (* y (* x (+ 1.0 (pow z 2.0)))))))))

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 t_0 = (1.0 / x) / (y * (1.0 + (z * z)));
	double tmp;
	if (t_0 <= -4.380498551296909e-285) {
		tmp = (1.0 / x) / ((y * hypot(1.0, z)) * sqrt(fma(z, z, 1.0)));
	} else if (t_0 <= 6.697308091806901e-18) {
		tmp = (((1.0 / x) / hypot(1.0, z)) / y) / hypot(1.0, z);
	} else {
		tmp = 1.0 / (y * (x * (1.0 + pow(z, 2.0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	5.3
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (/.f64 1 x) (*.f64 y (+.f64 1 (*.f64 z z)))) < -4.3804985512969091e-285

   1. Initial program 0.3

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

   2. Simplified 0.3

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

   3. Applied add-sqr-sqrt_binary64 0.3

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

   4. Applied associate-*r*_binary64 0.3

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

   5. Simplified 0.3

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

   if -4.3804985512969091e-285 < (/.f64 (/.f64 1 x) (*.f64 y (+.f64 1 (*.f64 z z)))) < 6.69730809180690081e-18

   1. Initial program 11.3

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

   2. Simplified 11.3

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

   3. Applied add-sqr-sqrt_binary64 11.3

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

   4. Applied associate-*r*_binary64 11.3

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

   5. Simplified 11.3

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

   6. Applied *-un-lft-identity_binary64 11.3

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

   7. Applied add-cube-cbrt_binary64 11.3

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

   8. Applied times-frac_binary64 11.3

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

   9. Applied times-frac_binary64 10.2

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

   10. Simplified 10.2

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

   11. Simplified 2.2

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

   12. Applied associate-*l/_binary64 2.1

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

   13. Simplified 2.1

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

   14. Applied associate-/r*_binary64 1.3

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

   if 6.69730809180690081e-18 < (/.f64 (/.f64 1 x) (*.f64 y (+.f64 1 (*.f64 z z))))

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   3. Applied add-sqr-sqrt_binary64 0.4

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

   4. Applied associate-*r*_binary64 0.4

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

   5. Simplified 0.4

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

   6. Taylor expanded in y around inf 0.4

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq -4.380498551296909 \cdot 10^{-285}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \sqrt{\mathsf{fma}\left(z, z, 1\right)}}\\ \mathbf{elif}\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)} \leq 6.697308091806901 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}}{y}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot \left(x \cdot \left(1 + {z}^{2}\right)\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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)))))