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

Average Error: 6.6 → 3.9

Time: 7.3s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -3.70384347346874 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{elif}\;y \leq 5.838039238768918 \cdot 10^{-31}:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\\ \frac{\frac{1}{x}}{t_0 \cdot t_0} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\frac{1}{x}}\\ \frac{t_1 \cdot t_1}{y} \cdot \left(\frac{\sqrt[3]{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\sqrt[3]{\frac{1}{\sqrt[3]{x}}}}{\mathsf{hypot}\left(1, z\right)}\right) \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.70384347346874e-310)
   (/ (/ 1.0 (* y x)) (fma z z 1.0))
   (if (<= y 5.838039238768918e-31)
     (let* ((t_0 (* (sqrt y) (hypot 1.0 z)))) (/ (/ 1.0 x) (* t_0 t_0)))
     (let* ((t_1 (cbrt (/ 1.0 x))))
       (*
        (/ (* t_1 t_1) y)
        (*
         (/ (cbrt (/ 1.0 (* (cbrt x) (cbrt x)))) (hypot 1.0 z))
         (/ (cbrt (/ 1.0 (cbrt 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 <= -3.70384347346874e-310) {
		tmp = (1.0 / (y * x)) / fma(z, z, 1.0);
	} else if (y <= 5.838039238768918e-31) {
		double t_0 = sqrt(y) * hypot(1.0, z);
		tmp = (1.0 / x) / (t_0 * t_0);
	} else {
		double t_1 = cbrt(1.0 / x);
		tmp = ((t_1 * t_1) / y) * ((cbrt(1.0 / (cbrt(x) * cbrt(x))) / hypot(1.0, z)) * (cbrt(1.0 / cbrt(x)) / hypot(1.0, z)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.6
Target	5.1
Herbie	3.9

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

Derivation

1. Split input into 3 regimes

2. if y < -3.703843473468744e-310

   1. Initial program 5.8

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

   2. Simplified 5.8

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

   3. Applied associate-/r*_binary64 7.5

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

   4. Simplified 7.7

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

   if -3.703843473468744e-310 < y < 5.83803923876891774e-31

   1. Initial program 11.0

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

   2. Simplified 11.0

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

   3. Applied add-sqr-sqrt_binary64 11.0

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

   4. Applied add-sqr-sqrt_binary64 11.2

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

   5. Applied unswap-sqr_binary64 11.2

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

   6. Simplified 11.2

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

   7. Simplified 4.5

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

   if 5.83803923876891774e-31 < y

   1. Initial program 5.0

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

   2. Simplified 5.0

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

   3. Applied add-cube-cbrt_binary64 5.5

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

   4. Applied times-frac_binary64 2.6

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

   5. Applied add-sqr-sqrt_binary64 2.6

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

   6. Applied add-cube-cbrt_binary64 2.6

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

   7. Applied add-cube-cbrt_binary64 2.6

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

   8. Applied times-frac_binary64 2.6

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

   9. Applied cbrt-prod_binary64 2.7

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

   10. Applied times-frac_binary64 2.7

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

   11. Simplified 2.7

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

   12. Simplified 1.7

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.70384347346874 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{1}{y \cdot x}}{\mathsf{fma}\left(z, z, 1\right)}\\ \mathbf{elif}\;y \leq 5.838039238768918 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\right) \cdot \left(\sqrt{y} \cdot \mathsf{hypot}\left(1, z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{y} \cdot \left(\frac{\sqrt[3]{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\mathsf{hypot}\left(1, z\right)} \cdot \frac{\sqrt[3]{\frac{1}{\sqrt[3]{x}}}}{\mathsf{hypot}\left(1, z\right)}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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)))))