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

Average Error: 15.1 → 1.5

Time: 6.7s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{1}{z}}{\frac{z \cdot \frac{z}{y}}{x}}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -3.2880636115797007 \cdot 10^{-102}:\\ \;\;\;\;\frac{{\left(\frac{\mathsf{fma}\left(z, z, z\right)}{x \cdot y}\right)}^{-1}}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.290464648854404 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;x \cdot y \leq 8.738469963482544 \cdot 10^{+280}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z + {z}^{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 z) (/ (* z (/ z y)) x))))
   (if (<= (* x y) (- INFINITY))
     t_0
     (if (<= (* x y) -3.2880636115797007e-102)
       (/ (pow (/ (fma z z z) (* x y)) -1.0) z)
       (if (<= (* x y) 3.290464648854404e-127)
         (* (/ x z) (/ y (fma z z z)))
         (if (<= (* x y) 8.738469963482544e+280)
           (/ (/ (* x y) (+ z (pow z 2.0))) z)
           t_0))))))

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 = (1.0 / z) / ((z * (z / y)) / x);
	double tmp;
	if ((x * y) <= -((double) INFINITY)) {
		tmp = t_0;
	} else if ((x * y) <= -3.2880636115797007e-102) {
		tmp = pow((fma(z, z, z) / (x * y)), -1.0) / z;
	} else if ((x * y) <= 3.290464648854404e-127) {
		tmp = (x / z) * (y / fma(z, z, z));
	} else if ((x * y) <= 8.738469963482544e+280) {
		tmp = ((x * y) / (z + pow(z, 2.0))) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	4.0
Herbie	1.5

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

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -inf.0 or 8.738469963482544e280 < (*.f64 x y)

   1. Initial program 59.9

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

   2. Simplified 20.2

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

   3. Applied egg-rr 22.5

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

   4. Applied egg-rr 22.5

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

   5. Applied egg-rr 21.9

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

   6. Taylor expanded in z around inf 22.3

      \[\leadsto \frac{\frac{1}{z}}{\frac{\color{blue}{\frac{{z}^{2}}{y}}}{x}} \]

   7. Simplified 5.3

      \[\leadsto \frac{\frac{1}{z}}{\frac{\color{blue}{\frac{z}{y} \cdot z}}{x}} \]

   if -inf.0 < (*.f64 x y) < -3.28806361157970073e-102

   1. Initial program 5.6

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

   2. Simplified 7.6

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

   3. Applied egg-rr 4.4

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

   4. Applied egg-rr 1.6

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

   if -3.28806361157970073e-102 < (*.f64 x y) < 3.290464648854404e-127

   1. Initial program 17.3

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

   2. Simplified 7.0

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

   3. Applied egg-rr 2.3

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

   4. Applied egg-rr 0.9

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

   if 3.290464648854404e-127 < (*.f64 x y) < 8.738469963482544e280

   1. Initial program 5.2

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

   2. Simplified 7.6

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

   3. Applied egg-rr 4.1

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

   4. Taylor expanded in x around 0 1.0

      \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{{z}^{2} + z}}}{z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{z \cdot \frac{z}{y}}{x}}\\ \mathbf{elif}\;x \cdot y \leq -3.2880636115797007 \cdot 10^{-102}:\\ \;\;\;\;\frac{{\left(\frac{\mathsf{fma}\left(z, z, z\right)}{x \cdot y}\right)}^{-1}}{z}\\ \mathbf{elif}\;x \cdot y \leq 3.290464648854404 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;x \cdot y \leq 8.738469963482544 \cdot 10^{+280}:\\ \;\;\;\;\frac{\frac{x \cdot y}{z + {z}^{2}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{z \cdot \frac{z}{y}}{x}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))