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

Average Error: 14.6 → 0.7

Time: 19.0s

Precision: binary64

Cost: 7756

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \frac{z + 1}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ y z) (/ z (/ x z)))))
   (if (<= (* x y) -2e+253)
     t_0
     (if (<= (* x y) -5e-245)
       (/ (/ (* x y) (fma z z z)) z)
       (if (<= (* x y) 2e-176)
         (/ (/ y z) (/ (fma z z z) x))
         (if (<= (* x y) 5e+300)
           (/ (/ 1.0 z) (* z (/ (+ z 1.0) (* x y))))
           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 = (y / z) / (z / (x / z));
	double tmp;
	if ((x * y) <= -2e+253) {
		tmp = t_0;
	} else if ((x * y) <= -5e-245) {
		tmp = ((x * y) / fma(z, z, z)) / z;
	} else if ((x * y) <= 2e-176) {
		tmp = (y / z) / (fma(z, z, z) / x);
	} else if ((x * y) <= 5e+300) {
		tmp = (1.0 / z) / (z * ((z + 1.0) / (x * y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

↓

function code(x, y, z)
	t_0 = Float64(Float64(y / z) / Float64(z / Float64(x / z)))
	tmp = 0.0
	if (Float64(x * y) <= -2e+253)
		tmp = t_0;
	elseif (Float64(x * y) <= -5e-245)
		tmp = Float64(Float64(Float64(x * y) / fma(z, z, z)) / z);
	elseif (Float64(x * y) <= 2e-176)
		tmp = Float64(Float64(y / z) / Float64(fma(z, z, z) / x));
	elseif (Float64(x * y) <= 5e+300)
		tmp = Float64(Float64(1.0 / z) / Float64(z * Float64(Float64(z + 1.0) / Float64(x * y))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] / N[(z / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+253], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], -5e-245], N[(N[(N[(x * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-176], N[(N[(y / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+300], N[(N[(1.0 / z), $MachinePrecision] / N[(z * N[(N[(z + 1.0), $MachinePrecision] / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Target

Original	14.6
Target	4.1
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 4 regimes

2. if (*.f64 x y) < -1.9999999999999999e253 or 5.00000000000000026e300 < (*.f64 x y)

   1. Initial program 54.2

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

   2. Simplified 18.2

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z (Rewrite<= *-lft-identity_binary64 (*.f64 1 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) (*.f64 1 z))))): 3 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 z 1))))): 2 points increase in error, 3 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 73 points increase in error, 23 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 z (*.f64 z (+.f64 z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (+.f64 z 1)))): 2 points increase in error, 2 points decrease in error

   3. Applied egg-rr 18.2

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

   4. Taylor expanded in z around inf 19.4

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

   5. Simplified 2.6

      \[\leadsto \frac{\frac{y}{z}}{\color{blue}{\frac{z}{\frac{x}{z}}}} \]
      Proof
      (/.f64 z (/.f64 x z)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 z z) x)): 61 points increase in error, 19 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) x): 0 points increase in error, 0 points decrease in error

   if -1.9999999999999999e253 < (*.f64 x y) < -4.9999999999999997e-245

   1. Initial program 6.1

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

   2. Simplified 5.2

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z (Rewrite<= *-lft-identity_binary64 (*.f64 1 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) (*.f64 1 z))))): 3 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 z 1))))): 2 points increase in error, 3 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 73 points increase in error, 23 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 z (*.f64 z (+.f64 z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (+.f64 z 1)))): 2 points increase in error, 2 points decrease in error

   3. Applied egg-rr 0.6

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

   if -4.9999999999999997e-245 < (*.f64 x y) < 2e-176

   1. Initial program 20.6

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]
      Proof
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z z))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (fma.f64 z z (Rewrite<= *-lft-identity_binary64 (*.f64 1 z))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) (*.f64 1 z))))): 3 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y z) (/.f64 x (Rewrite<= distribute-rgt-in_binary64 (*.f64 z (+.f64 z 1))))): 2 points increase in error, 3 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y x) (*.f64 z (*.f64 z (+.f64 z 1))))): 73 points increase in error, 23 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x y)) (*.f64 z (*.f64 z (+.f64 z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 x y) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 z z) (+.f64 z 1)))): 2 points increase in error, 2 points decrease in error

   3. Applied egg-rr 0.5

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

   if 2e-176 < (*.f64 x y) < 5.00000000000000026e300

   1. Initial program 6.3

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

   2. Applied egg-rr 3.5

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

   3. Applied egg-rr 3.6

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

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{\frac{1}{z}}{\frac{1 + z}{x \cdot y} \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-176}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \frac{z + 1}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	7496

\[\begin{array}{l} t_0 := \frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-303}:\\ \;\;\;\;\frac{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{elif}\;x \cdot y \leq 10^{-310}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{1}{z}}{z \cdot \frac{z + 1}{x \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1872

\[\begin{array}{l} t_0 := \frac{\frac{1}{z}}{z \cdot \frac{z + 1}{x \cdot y}}\\ t_1 := \frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{if}\;x \cdot y \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 10^{-310}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	3.5
Cost	1744

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{z + 1}{y}}}{z \cdot z}\\ t_1 := \frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 10^{-201}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{+193}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	18.7
Cost	1488

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+15}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{elif}\;x \cdot y \leq -2 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 5
Error	2.9
Cost	1224

\[\begin{array}{l} t_0 := \frac{z + 1}{y}\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \mathbf{elif}\;x \cdot y \leq -1 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot t_0}}{z}\\ \end{array} \]

Alternative 6
Error	6.5
Cost	840

\[\begin{array}{l} t_0 := \frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	4.7
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-10}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	4.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{\frac{y}{\frac{z}{x}}}{z}}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	712

\[\begin{array}{l} t_0 := x \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 10^{-51}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	18.3
Cost	580

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

Alternative 11
Error	18.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \end{array} \]

Alternative 12
Error	43.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+54}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 13
Error	42.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 6.725307621755194 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 14
Error	42.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 6.725307621755194 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 15
Error	22.2
Cost	448

\[\frac{\frac{y}{\frac{z}{x}}}{z} \]

Alternative 16
Error	48.9
Cost	320

\[\frac{x \cdot y}{z} \]

Alternative 17
Error	46.1
Cost	320

\[x \cdot \frac{y}{z} \]

Reproduce

herbie shell --seed 2022317 
(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))))