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

Average Error: 14.9 → 3.8

Time: 19.8s

Precision: binary64

Cost: 1232

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x y) (* (* z z) (+ z 1.0)))))
   (if (<= z -3.3e+92)
     (* (/ y z) (/ (/ x z) (+ z 1.0)))
     (if (<= z -2.4e-108)
       t_0
       (if (<= z 3e-129)
         (* (/ y z) (/ x z))
         (if (<= z 8e+66) t_0 (/ (/ x (/ z (/ y z))) 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 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (z <= -3.3e+92) {
		tmp = (y / z) * ((x / z) / (z + 1.0));
	} else if (z <= -2.4e-108) {
		tmp = t_0;
	} else if (z <= 3e-129) {
		tmp = (y / z) * (x / z);
	} else if (z <= 8e+66) {
		tmp = t_0;
	} else {
		tmp = (x / (z / (y / z))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * y) / ((z * z) * (z + 1.0d0))
    if (z <= (-3.3d+92)) then
        tmp = (y / z) * ((x / z) / (z + 1.0d0))
    else if (z <= (-2.4d-108)) then
        tmp = t_0
    else if (z <= 3d-129) then
        tmp = (y / z) * (x / z)
    else if (z <= 8d+66) then
        tmp = t_0
    else
        tmp = (x / (z / (y / z))) / z
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (z <= -3.3e+92) {
		tmp = (y / z) * ((x / z) / (z + 1.0));
	} else if (z <= -2.4e-108) {
		tmp = t_0;
	} else if (z <= 3e-129) {
		tmp = (y / z) * (x / z);
	} else if (z <= 8e+66) {
		tmp = t_0;
	} else {
		tmp = (x / (z / (y / z))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

↓

def code(x, y, z):
	t_0 = (x * y) / ((z * z) * (z + 1.0))
	tmp = 0
	if z <= -3.3e+92:
		tmp = (y / z) * ((x / z) / (z + 1.0))
	elif z <= -2.4e-108:
		tmp = t_0
	elif z <= 3e-129:
		tmp = (y / z) * (x / z)
	elif z <= 8e+66:
		tmp = t_0
	else:
		tmp = (x / (z / (y / z))) / z
	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(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
	tmp = 0.0
	if (z <= -3.3e+92)
		tmp = Float64(Float64(y / z) * Float64(Float64(x / z) / Float64(z + 1.0)));
	elseif (z <= -2.4e-108)
		tmp = t_0;
	elseif (z <= 3e-129)
		tmp = Float64(Float64(y / z) * Float64(x / z));
	elseif (z <= 8e+66)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(z / Float64(y / z))) / z);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * y) / ((z * z) * (z + 1.0));
	tmp = 0.0;
	if (z <= -3.3e+92)
		tmp = (y / z) * ((x / z) / (z + 1.0));
	elseif (z <= -2.4e-108)
		tmp = t_0;
	elseif (z <= 3e-129)
		tmp = (y / z) * (x / z);
	elseif (z <= 8e+66)
		tmp = t_0;
	else
		tmp = (x / (z / (y / z))) / z;
	end
	tmp_2 = 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[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+92], N[(N[(y / z), $MachinePrecision] * N[(N[(x / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.4e-108], t$95$0, If[LessEqual[z, 3e-129], N[(N[(y / z), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e+66], t$95$0, N[(N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]]

Target

Original	14.9
Target	4.1
Herbie	3.8

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

Derivation

1. Split input into 4 regimes

2. if z < -3.29999999999999974e92

   1. Initial program 12.5

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

   2. Simplified 4.8

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

      [Start] 12.5	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      rational.json-simplify-46 [=>] 8.6	\[ \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]
      rational.json-simplify-2 [=>] 8.6	\[ \frac{\frac{\color{blue}{y \cdot x}}{z \cdot z}}{z + 1} \]
      rational.json-simplify-49 [=>] 4.8	\[ \frac{\color{blue}{x \cdot \frac{y}{z \cdot z}}}{z + 1} \]

   3. Applied egg-rr 1.5

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

   4. Simplified 1.5

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

      [Start] 1.5	\[ \frac{y}{z} \cdot \frac{\frac{1}{z}}{\frac{z + 1}{x}} \]
      rational.json-simplify-44 [=>] 1.5	\[ \frac{y}{z} \cdot \color{blue}{\frac{\frac{1}{\frac{z + 1}{x}}}{z}} \]
      rational.json-simplify-61 [=>] 1.5	\[ \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{\frac{z + 1}{1}}}}{z} \]
      rational.json-simplify-7 [=>] 1.5	\[ \frac{y}{z} \cdot \frac{\frac{x}{\color{blue}{z + 1}}}{z} \]
      rational.json-simplify-44 [=>] 1.5	\[ \frac{y}{z} \cdot \color{blue}{\frac{\frac{x}{z}}{z + 1}} \]

   if -3.29999999999999974e92 < z < -2.40000000000000017e-108 or 2.9999999999999998e-129 < z < 7.99999999999999956e66

   1. Initial program 5.6

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

   if -2.40000000000000017e-108 < z < 2.9999999999999998e-129

   1. Initial program 46.4

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

   2. Simplified 46.7

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

      [Start] 46.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      rational.json-simplify-46 [=>] 46.4	\[ \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}} \]
      rational.json-simplify-2 [=>] 46.4	\[ \frac{\frac{\color{blue}{y \cdot x}}{z \cdot z}}{z + 1} \]
      rational.json-simplify-49 [=>] 46.7	\[ \frac{\color{blue}{x \cdot \frac{y}{z \cdot z}}}{z + 1} \]

   3. Applied egg-rr 6.5

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

   4. Simplified 6.4

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

      [Start] 6.5	\[ \frac{y}{z} \cdot \frac{\frac{1}{z}}{\frac{z + 1}{x}} \]
      rational.json-simplify-44 [=>] 6.5	\[ \frac{y}{z} \cdot \color{blue}{\frac{\frac{1}{\frac{z + 1}{x}}}{z}} \]
      rational.json-simplify-61 [=>] 6.4	\[ \frac{y}{z} \cdot \frac{\color{blue}{\frac{x}{\frac{z + 1}{1}}}}{z} \]
      rational.json-simplify-7 [=>] 6.4	\[ \frac{y}{z} \cdot \frac{\frac{x}{\color{blue}{z + 1}}}{z} \]
      rational.json-simplify-44 [=>] 6.4	\[ \frac{y}{z} \cdot \color{blue}{\frac{\frac{x}{z}}{z + 1}} \]

   5. Taylor expanded in z around 0 6.4

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

   if 7.99999999999999956e66 < z

   1. Initial program 12.0

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

   2. Simplified 6.0

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

      [Start] 12.0	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      rational.json-simplify-49 [=>] 9.3	\[ \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]
      rational.json-simplify-46 [=>] 6.0	\[ y \cdot \color{blue}{\frac{\frac{x}{z \cdot z}}{z + 1}} \]

   3. Applied egg-rr 1.7

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}} + 0} \]

   4. Simplified 1.9

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

      [Start] 1.7	\[ \frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}} + 0 \]
      rational.json-simplify-4 [=>] 1.7	\[ \color{blue}{\frac{\frac{y}{z}}{z \cdot \frac{z + 1}{x}}} \]
      rational.json-simplify-44 [=>] 1.8	\[ \color{blue}{\frac{\frac{y}{z \cdot \frac{z + 1}{x}}}{z}} \]
      rational.json-simplify-46 [=>] 0.8	\[ \frac{\color{blue}{\frac{\frac{y}{z}}{\frac{z + 1}{x}}}}{z} \]
      rational.json-simplify-61 [<=] 1.9	\[ \frac{\color{blue}{\frac{x}{\frac{z + 1}{\frac{y}{z}}}}}{z} \]
      rational.json-simplify-61 [<=] 1.9	\[ \frac{\frac{x}{\color{blue}{\frac{z}{\frac{y}{z + 1}}}}}{z} \]

   5. Taylor expanded in z around inf 1.9

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

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+66}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.5
Cost	1736

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ t_1 := \frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-41}:\\ \;\;\;\;\frac{\frac{y \cdot x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	3.1
Cost	1232

\[\begin{array}{l} t_0 := y \cdot \frac{\frac{x}{z \cdot z}}{z + 1}\\ t_1 := \frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;z \leq 80000000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	4.1
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{if}\;z \leq -2.45 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-136}:\\ \;\;\;\;x \cdot \frac{y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	3.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\frac{x}{z}}{z + 1}\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z + 1}}}}{z}\\ \end{array} \]

Alternative 5
Error	4.4
Cost	840

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

Alternative 6
Error	17.9
Cost	712

\[\begin{array}{l} t_0 := \frac{x}{z \cdot z} \cdot y\\ \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	17.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-295}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]

Alternative 8
Error	17.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]

Alternative 9
Error	17.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{x}}\\ \end{array} \]

Alternative 10
Error	43.4
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{z} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \]

Alternative 11
Error	43.0
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \end{array} \]

Alternative 12
Error	43.2
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+102}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 13
Error	24.0
Cost	448

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

Alternative 14
Error	46.4
Cost	384

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

Reproduce

herbie shell --seed 2023074 
(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))))