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

Average Error: 6.7 → 5.4

Time: 5.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -9.89983358260154139 \cdot 10^{51}:\\ \;\;\;\;\frac{\frac{1}{1}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{x} \le 3.57512253180957601 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\\ \end{array}\]

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 temp;
	if (((1.0 / x) <= -9.899833582601541e+51)) {
		temp = ((1.0 / 1.0) / (y * (sqrt(fma(z, z, 1.0)) * (sqrt(fma(z, z, 1.0)) * x))));
	} else {
		double temp_1;
		if (((1.0 / x) <= 3.575122531809576e+101)) {
			temp_1 = ((1.0 / x) / (y * (1.0 + (z * z))));
		} else {
			temp_1 = ((1.0 / 1.0) * ((1.0 / y) / (fma(z, z, 1.0) * x)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.7
Target	6.1
Herbie	5.4

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/ 1.0 x) < -9.899833582601541e+51

   1. Initial program 13.9

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

   2. Simplified 10.4

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

   4. Applied *-un-lft-identity 10.4

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

   5. Applied div-inv 10.4

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

   6. Applied times-frac 10.4

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

   7. Applied associate-/l* 10.5

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

   8. Simplified 10.3

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

   10. Applied add-sqr-sqrt 10.3

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

   11. Applied associate-*l* 10.3

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

   if -9.899833582601541e+51 < (/ 1.0 x) < 3.575122531809576e+101

   1. Initial program 2.7

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

   if 3.575122531809576e+101 < (/ 1.0 x)

   1. Initial program 16.2

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

   2. Simplified 12.1

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

   4. Applied *-un-lft-identity 12.1

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

   5. Applied div-inv 12.1

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

   6. Applied times-frac 12.1

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

   7. Applied associate-/l* 12.2

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

   8. Simplified 12.1

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

   10. Applied add-sqr-sqrt 12.1

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

   11. Applied associate-*l* 12.1

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

   13. Applied div-inv 12.1

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

   14. Simplified 12.1

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

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x} \le -9.89983358260154139 \cdot 10^{51}:\\ \;\;\;\;\frac{\frac{1}{1}}{y \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot \left(\sqrt{\mathsf{fma}\left(z, z, 1\right)} \cdot x\right)\right)}\\ \mathbf{elif}\;\frac{1}{x} \le 3.57512253180957601 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1} \cdot \frac{\frac{1}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))