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

Average Error: 6.8 → 1.3

Time: 8.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5355777734533977 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5355777734533977e-73)
   (* (/ (/ 1.0 y) (hypot 1.0 z)) (/ (/ 1.0 x) (hypot 1.0 z)))
   (* (/ 1.0 (hypot 1.0 z)) (/ (/ 1.0 (* y 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 (y <= 4.5355777734533977e-73) {
		tmp = ((1.0 / y) / hypot(1.0, z)) * ((1.0 / x) / hypot(1.0, z));
	} else {
		tmp = (1.0 / hypot(1.0, z)) * ((1.0 / (y * x)) / hypot(1.0, z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.8
Target	5.4
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if y < 4.53557777345339767e-73

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   3. Applied *-un-lft-identity_binary64 9.0

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

   4. Applied add-cube-cbrt_binary64 9.0

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

   5. Applied times-frac_binary64 9.0

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

   6. Applied times-frac_binary64 11.7

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

   7. Simplified 11.7

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

   8. Simplified 11.9

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

   9. Applied un-div-inv_binary64 11.9

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

   10. Applied associate-/r*_binary64 12.3

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

   11. Simplified 12.4

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

   12. Applied add-sqr-sqrt_binary64 12.4

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

   13. Applied add-cube-cbrt_binary64 12.4

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

   14. Applied times-frac_binary64 12.4

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

   15. Applied times-frac_binary64 10.1

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

   16. Simplified 10.1

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

   17. Simplified 1.7

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

   if 4.53557777345339767e-73 < y

   1. Initial program 5.1

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

   2. Simplified 5.1

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

   3. Applied *-un-lft-identity_binary64 5.1

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

   4. Applied add-cube-cbrt_binary64 5.1

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

   5. Applied times-frac_binary64 5.1

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

   6. Applied times-frac_binary64 2.7

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

   7. Simplified 2.7

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

   8. Simplified 2.7

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

   9. Applied un-div-inv_binary64 2.7

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

   10. Applied associate-/r*_binary64 2.8

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

   11. Simplified 3.1

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

   12. Applied add-sqr-sqrt_binary64 3.1

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

   13. Applied *-un-lft-identity_binary64 3.1

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

   14. Applied times-frac_binary64 3.1

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

   15. Simplified 3.1

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

   16. Simplified 0.9

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5355777734533977 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{1}{y}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\frac{1}{y \cdot x}}{\mathsf{hypot}\left(1, z\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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)))))