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

Average Error: 6.7 → 0.5

Time: 10.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4.593935847763388 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{\sqrt{y}}\right) \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\sqrt{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* y (+ 1.0 (* z z))) 4.593935847763388e+307)
   (/ (/ 1.0 x) (* (hypot 1.0 z) (* y (hypot 1.0 z))))
   (*
    (* (/ (/ 1.0 x) (hypot 1.0 z)) (/ 1.0 (sqrt y)))
    (/ (/ 1.0 (hypot 1.0 z)) (sqrt y)))))

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 * (1.0 + (z * z))) <= 4.593935847763388e+307) {
		tmp = (1.0 / x) / (hypot(1.0, z) * (y * hypot(1.0, z)));
	} else {
		tmp = (((1.0 / x) / hypot(1.0, z)) * (1.0 / sqrt(y))) * ((1.0 / hypot(1.0, z)) / sqrt(y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	5.2
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (+.f64 1 (*.f64 z z))) < 4.5939358477633883e307

   1. Initial program 2.1

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

   2. Simplified 2.1

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

   3. Applied associate-/r*_binary64 4.3

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

   4. Applied add-sqr-sqrt_binary64 4.3

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

   5. Applied div-inv_binary64 4.4

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

   6. Applied times-frac_binary64 2.6

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

   7. Simplified 2.6

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

   8. Simplified 1.0

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

   9. Applied pow1_binary64 1.0

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

   10. Applied pow1_binary64 1.0

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

   11. Applied pow-prod-down_binary64 1.0

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

   12. Simplified 0.5

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

   if 4.5939358477633883e307 < (*.f64 y (+.f64 1 (*.f64 z z)))

   1. Initial program 19.2

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

   2. Simplified 19.2

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

   3. Applied associate-/r*_binary64 13.9

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

   4. Applied add-sqr-sqrt_binary64 13.9

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

   5. Applied div-inv_binary64 13.9

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

   6. Applied times-frac_binary64 15.4

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

   7. Simplified 15.4

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

   8. Simplified 3.1

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

   9. Applied add-sqr-sqrt_binary64 3.1

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

   10. Applied *-un-lft-identity_binary64 3.1

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

   11. Applied times-frac_binary64 3.1

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

   12. Applied associate-*r*_binary64 0.6

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \leq 4.593935847763388 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right) \cdot \left(y \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{x}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{1}{\sqrt{y}}\right) \cdot \frac{\frac{1}{\mathsf{hypot}\left(1, z\right)}}{\sqrt{y}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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)))))