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

Average Error: 22.54% → 6.6%

Time: 12.0s

Precision: binary64

Cost: 2768

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ \mathbf{if}\;t_0 \leq -1000000:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;t_0 \leq 10^{-161}:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}\\ \mathbf{elif}\;t_0 \leq 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (if (<= t_0 -1000000.0)
     (/ (/ x (/ z (/ y z))) z)
     (if (<= t_0 2e-278)
       (/ (/ x (/ z y)) z)
       (if (<= t_0 1e-161)
         (/ x (* z (* z (/ (+ z 1.0) y))))
         (if (<= t_0 1e+238) (/ (* x y) t_0) (* (/ (/ x z) z) (/ y 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 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (x / (z / (y / z))) / z;
	} else if (t_0 <= 2e-278) {
		tmp = (x / (z / y)) / z;
	} else if (t_0 <= 1e-161) {
		tmp = x / (z * (z * ((z + 1.0) / y)));
	} else if (t_0 <= 1e+238) {
		tmp = (x * y) / t_0;
	} else {
		tmp = ((x / z) / z) * (y / 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 = (z * z) * (z + 1.0d0)
    if (t_0 <= (-1000000.0d0)) then
        tmp = (x / (z / (y / z))) / z
    else if (t_0 <= 2d-278) then
        tmp = (x / (z / y)) / z
    else if (t_0 <= 1d-161) then
        tmp = x / (z * (z * ((z + 1.0d0) / y)))
    else if (t_0 <= 1d+238) then
        tmp = (x * y) / t_0
    else
        tmp = ((x / z) / z) * (y / 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 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1000000.0) {
		tmp = (x / (z / (y / z))) / z;
	} else if (t_0 <= 2e-278) {
		tmp = (x / (z / y)) / z;
	} else if (t_0 <= 1e-161) {
		tmp = x / (z * (z * ((z + 1.0) / y)));
	} else if (t_0 <= 1e+238) {
		tmp = (x * y) / t_0;
	} else {
		tmp = ((x / z) / z) * (y / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if t_0 <= -1000000.0:
		tmp = (x / (z / (y / z))) / z
	elif t_0 <= 2e-278:
		tmp = (x / (z / y)) / z
	elif t_0 <= 1e-161:
		tmp = x / (z * (z * ((z + 1.0) / y)))
	elif t_0 <= 1e+238:
		tmp = (x * y) / t_0
	else:
		tmp = ((x / z) / z) * (y / 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(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -1000000.0)
		tmp = Float64(Float64(x / Float64(z / Float64(y / z))) / z);
	elseif (t_0 <= 2e-278)
		tmp = Float64(Float64(x / Float64(z / y)) / z);
	elseif (t_0 <= 1e-161)
		tmp = Float64(x / Float64(z * Float64(z * Float64(Float64(z + 1.0) / y))));
	elseif (t_0 <= 1e+238)
		tmp = Float64(Float64(x * y) / t_0);
	else
		tmp = Float64(Float64(Float64(x / z) / z) * Float64(y / 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 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_0 <= -1000000.0)
		tmp = (x / (z / (y / z))) / z;
	elseif (t_0 <= 2e-278)
		tmp = (x / (z / y)) / z;
	elseif (t_0 <= 1e-161)
		tmp = x / (z * (z * ((z + 1.0) / y)));
	elseif (t_0 <= 1e+238)
		tmp = (x * y) / t_0;
	else
		tmp = ((x / z) / z) * (y / 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[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000000.0], N[(N[(x / N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2e-278], N[(N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-161], N[(x / N[(z * N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+238], N[(N[(x * y), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	22.54%
Target	6.31%
Herbie	6.6%

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

Derivation

1. Split input into 5 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < -1e6

   1. Initial program 17.12

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

   2. Simplified 8.08

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

      [Start] 17.12	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 8.08	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.93

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

   4. Taylor expanded in z around inf 10.27

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

   5. Simplified 6.15

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

      [Start] 10.27	\[ \frac{\frac{x}{\frac{{z}^{2}}{y}}}{z} \]
      unpow2 [=>] 10.27	\[ \frac{\frac{x}{\frac{\color{blue}{z \cdot z}}{y}}}{z} \]
      associate-/l* [=>] 6.15	\[ \frac{\frac{x}{\color{blue}{\frac{z}{\frac{y}{z}}}}}{z} \]

   if -1e6 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.99999999999999988e-278

   1. Initial program 82.37

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

   2. Simplified 84.03

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

      [Start] 82.37	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 84.03	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 7.93

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

   4. Taylor expanded in z around 0 9.32

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

   if 1.99999999999999988e-278 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.00000000000000003e-161

   1. Initial program 16.89

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

   2. Simplified 17.92

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

      [Start] 16.89	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 17.92	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 15.56

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

   if 1.00000000000000003e-161 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1e238

   1. Initial program 7.14

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

   if 1e238 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 18.62

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

   2. Simplified 7.25

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

      [Start] 18.62	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 7.25	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 2.12

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

   4. Applied egg-rr 2.32

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

   5. Taylor expanded in z around inf 2.32

      \[\leadsto \frac{\frac{x}{z}}{z} \cdot \color{blue}{\frac{y}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1000000:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-161}:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{+238}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.13%
Cost	2249

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-299} \lor \neg \left(t_0 \leq 10^{-263}\right):\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \end{array} \]

Alternative 2
Error	4.18%
Cost	2248

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{x \cdot y}{z + z \cdot z}\\ \mathbf{elif}\;t_0 \leq 10^{-263}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \end{array} \]

Alternative 3
Error	5.01%
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-115} \lor \neg \left(z \leq 3.2 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 4
Error	7.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 5
Error	9.79%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 6
Error	7.04%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 7
Error	7.45%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;\frac{\frac{x}{\frac{z}{\frac{y}{z}}}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 8
Error	6.97%
Cost	840

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

Alternative 9
Error	28.02%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{-54} \lor \neg \left(z \leq 5 \cdot 10^{-41}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 10
Error	27.91%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{-109} \lor \neg \left(z \leq 2.9 \cdot 10^{-134}\right):\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \end{array} \]

Alternative 11
Error	27.96%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-137} \lor \neg \left(z \leq 10^{-131}\right):\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 12
Error	26.73%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 13
Error	26.52%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{\frac{z \cdot z}{y}}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-186}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 14
Error	29.04%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 15
Error	66.97%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \end{array} \]

Alternative 16
Error	34.62%
Cost	448

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

Alternative 17
Error	72.41%
Cost	384

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

Alternative 18
Error	71.9%
Cost	384

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

Reproduce

herbie shell --seed 2023088 
(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))))