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

Average Error: 15.1 → 0.7

Time: 5.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\frac{z}{x} \cdot \left(\frac{z}{y} + \frac{z}{\frac{y}{z}}\right)}\\ \mathbf{if}\;x \cdot y \leq -1.6594710605531548 \cdot 10^{+205}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{if}\;x \cdot y \leq -2.275563410703594 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq 7.7566315 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}}\\ \mathbf{elif}\;x \cdot y \leq 1.5254635472042655 \cdot 10^{+188}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (/ z x) (+ (/ z y) (/ z (/ y z)))))))
   (if (<= (* x y) -1.6594710605531548e+205)
     t_0
     (let* ((t_1 (/ (/ (* x y) (fma z z z)) z)))
       (if (<= (* x y) -2.275563410703594e-152)
         t_1
         (if (<= (* x y) 7.7566315e-316)
           (/ 1.0 (/ (/ z x) (/ y (fma z z z))))
           (if (<= (* x y) 1.5254635472042655e+188) t_1 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 / x) * ((z / y) + (z / (y / z))));
	double tmp;
	if ((x * y) <= -1.6594710605531548e+205) {
		tmp = t_0;
	} else {
		double t_1 = ((x * y) / fma(z, z, z)) / z;
		double tmp_1;
		if ((x * y) <= -2.275563410703594e-152) {
			tmp_1 = t_1;
		} else if ((x * y) <= 7.7566315e-316) {
			tmp_1 = 1.0 / ((z / x) / (y / fma(z, z, z)));
		} else if ((x * y) <= 1.5254635472042655e+188) {
			tmp_1 = t_1;
		} else {
			tmp_1 = t_0;
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	4.0
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.65947106055315481e205 or 1.5254635472042655e188 < (*.f64 x y)

   1. Initial program 38.1

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

   2. Simplified 13.9

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

   4. Applied *-commutative_binary64 13.9

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

   5. Applied associate-*l/_binary64 14.0

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

   6. Simplified 14.0

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

   8. Applied add-cube-cbrt_binary64 14.4

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

   9. Applied associate-*l*_binary64 14.4

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

   10. Simplified 14.4

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

   12. Applied clear-num_binary64 14.4

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

   13. Simplified 14.0

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

   14. Taylor expanded around 0 43.7

       \[\leadsto \frac{1}{\color{blue}{\frac{{z}^{3}}{y \cdot x} + \frac{{z}^{2}}{y \cdot x}}} \]

   15. Simplified 2.0

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

   if -1.65947106055315481e205 < (*.f64 x y) < -2.2755634107035938e-152 or 7.756631482e-316 < (*.f64 x y) < 1.5254635472042655e188

   1. Initial program 6.2

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

   2. Simplified 7.5

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

   4. Applied *-commutative_binary64 7.5

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

   5. Applied associate-*l/_binary64 2.9

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

   6. Simplified 2.9

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

   8. Applied pow1_binary64 2.9

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

   9. Applied pow1_binary64 2.9

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

   10. Applied pow-prod-down_binary64 2.9

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

   11. Simplified 0.4

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

   if -2.2755634107035938e-152 < (*.f64 x y) < 7.756631482e-316

   1. Initial program 21.0

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

   2. Simplified 6.4

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

   4. Applied *-commutative_binary64 6.4

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

   5. Applied associate-*l/_binary64 2.7

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

   6. Simplified 2.7

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

   8. Applied add-cube-cbrt_binary64 3.1

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

   9. Applied associate-*l*_binary64 3.2

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

   10. Simplified 3.2

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

   12. Applied clear-num_binary64 3.3

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

   13. Simplified 2.9

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

   15. Applied add-cube-cbrt_binary64 3.3

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

   16. Applied times-frac_binary64 3.7

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

   17. Applied associate-*l/_binary64 1.2

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

   18. Simplified 0.7

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.6594710605531548 \cdot 10^{+205}:\\ \;\;\;\;\frac{1}{\frac{z}{x} \cdot \left(\frac{z}{y} + \frac{z}{\frac{y}{z}}\right)}\\ \mathbf{elif}\;x \cdot y \leq -2.275563410703594 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x \cdot y \leq 7.7566315 \cdot 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{\frac{z}{x}}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}}\\ \mathbf{elif}\;x \cdot y \leq 1.5254635472042655 \cdot 10^{+188}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x} \cdot \left(\frac{z}{y} + \frac{z}{\frac{y}{z}}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(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))))