Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 15.4 → 2.9

Time: 5.9s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{y} \cdot \sqrt[3]{y}\\ \mathbf{if}\;z \leq -5.332363795460978 \cdot 10^{-308}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}\\ \left(x \cdot \frac{t_0}{t_1 \cdot t_1}\right) \cdot \frac{\frac{\sqrt[3]{y}}{t_1}}{z} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \mathsf{hypot}\left(z, \sqrt{z}\right)\\ \frac{t_0}{t_2} \cdot \left(\frac{\sqrt[3]{y}}{t_2} \cdot \frac{x}{z}\right) \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cbrt y) (cbrt y))))
   (if (<= z -5.332363795460978e-308)
     (let* ((t_1 (cbrt (fma z z z))))
       (* (* x (/ t_0 (* t_1 t_1))) (/ (/ (cbrt y) t_1) z)))
     (let* ((t_2 (hypot z (sqrt z))))
       (* (/ t_0 t_2) (* (/ (cbrt y) t_2) (/ x z)))))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

double code(double x, double y, double z) {
	double t_0 = cbrt(y) * cbrt(y);
	double tmp;
	if (z <= -5.332363795460978e-308) {
		double t_1_1 = cbrt(fma(z, z, z));
		tmp = (x * (t_0 / (t_1_1 * t_1_1))) * ((cbrt(y) / t_1_1) / z);
	} else {
		double t_2 = hypot(z, sqrt(z));
		tmp = (t_0 / t_2) * ((cbrt(y) / t_2) * (x / z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.4
Target	4.0
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -5.3323637954609783e-308

   1. Initial program 15.2

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

   2. Simplified 8.0

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

   3. Applied *-un-lft-identity_binary64 8.0

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

   4. Applied add-cube-cbrt_binary64 8.4

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

   5. Applied add-cube-cbrt_binary64 8.5

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

   6. Applied times-frac_binary64 8.5

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

   7. Applied times-frac_binary64 8.5

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

   8. Applied associate-*r*_binary64 3.8

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

   9. Simplified 3.8

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

   if -5.3323637954609783e-308 < z

   1. Initial program 15.6

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

   2. Simplified 8.9

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

   3. Applied associate-*r/_binary64 4.8

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

   4. Applied associate-/l*_binary64 8.7

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

   5. Applied add-sqr-sqrt_binary64 8.8

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

   6. Applied add-cube-cbrt_binary64 9.1

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

   7. Applied times-frac_binary64 9.1

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

   8. Applied *-un-lft-identity_binary64 9.1

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

   9. Applied times-frac_binary64 9.3

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

   10. Applied *-un-lft-identity_binary64 9.3

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

   11. Applied times-frac_binary64 5.6

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

   12. Simplified 5.6

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

   13. Simplified 2.0

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.332363795460978 \cdot 10^{-308}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)} \cdot \sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}\right) \cdot \frac{\frac{\sqrt[3]{y}}{\sqrt[3]{\mathsf{fma}\left(z, z, z\right)}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\mathsf{hypot}\left(z, \sqrt{z}\right)} \cdot \left(\frac{\sqrt[3]{y}}{\mathsf{hypot}\left(z, \sqrt{z}\right)} \cdot \frac{x}{z}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))