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

Average Accuracy: 77.2% → 97.1%

Time: 12.8s

Precision: binary64

Cost: 1872

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{-\frac{x \cdot y}{z + 1}}{z \cdot \left(-z\right)}\\ t_1 := \frac{-1}{\left(z \cdot \frac{z}{x}\right) \cdot \frac{-1 - z}{y}}\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 1.0))) (* z (- z))))
        (t_1 (/ -1.0 (* (* z (/ z x)) (/ (- -1.0 z) y)))))
   (if (<= (* x y) -5e+274)
     t_1
     (if (<= (* x y) -5e-162)
       t_0
       (if (<= (* x y) 4e-141)
         (/ (/ x z) (* z (/ (+ z 1.0) y)))
         (if (<= (* x y) 1e+176) t_0 t_1))))))

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 + 1.0)) / (z * -z);
	double t_1 = -1.0 / ((z * (z / x)) * ((-1.0 - z) / y));
	double tmp;
	if ((x * y) <= -5e+274) {
		tmp = t_1;
	} else if ((x * y) <= -5e-162) {
		tmp = t_0;
	} else if ((x * y) <= 4e-141) {
		tmp = (x / z) / (z * ((z + 1.0) / y));
	} else if ((x * y) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = -((x * y) / (z + 1.0d0)) / (z * -z)
    t_1 = (-1.0d0) / ((z * (z / x)) * (((-1.0d0) - z) / y))
    if ((x * y) <= (-5d+274)) then
        tmp = t_1
    else if ((x * y) <= (-5d-162)) then
        tmp = t_0
    else if ((x * y) <= 4d-141) then
        tmp = (x / z) / (z * ((z + 1.0d0) / y))
    else if ((x * y) <= 1d+176) then
        tmp = t_0
    else
        tmp = t_1
    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 + 1.0)) / (z * -z);
	double t_1 = -1.0 / ((z * (z / x)) * ((-1.0 - z) / y));
	double tmp;
	if ((x * y) <= -5e+274) {
		tmp = t_1;
	} else if ((x * y) <= -5e-162) {
		tmp = t_0;
	} else if ((x * y) <= 4e-141) {
		tmp = (x / z) / (z * ((z + 1.0) / y));
	} else if ((x * y) <= 1e+176) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 + 1.0)) / (z * -z)
	t_1 = -1.0 / ((z * (z / x)) * ((-1.0 - z) / y))
	tmp = 0
	if (x * y) <= -5e+274:
		tmp = t_1
	elif (x * y) <= -5e-162:
		tmp = t_0
	elif (x * y) <= 4e-141:
		tmp = (x / z) / (z * ((z + 1.0) / y))
	elif (x * y) <= 1e+176:
		tmp = t_0
	else:
		tmp = t_1
	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(-Float64(Float64(x * y) / Float64(z + 1.0))) / Float64(z * Float64(-z)))
	t_1 = Float64(-1.0 / Float64(Float64(z * Float64(z / x)) * Float64(Float64(-1.0 - z) / y)))
	tmp = 0.0
	if (Float64(x * y) <= -5e+274)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e-162)
		tmp = t_0;
	elseif (Float64(x * y) <= 4e-141)
		tmp = Float64(Float64(x / z) / Float64(z * Float64(Float64(z + 1.0) / y)));
	elseif (Float64(x * y) <= 1e+176)
		tmp = t_0;
	else
		tmp = t_1;
	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 + 1.0)) / (z * -z);
	t_1 = -1.0 / ((z * (z / x)) * ((-1.0 - z) / y));
	tmp = 0.0;
	if ((x * y) <= -5e+274)
		tmp = t_1;
	elseif ((x * y) <= -5e-162)
		tmp = t_0;
	elseif ((x * y) <= 4e-141)
		tmp = (x / z) / (z * ((z + 1.0) / y));
	elseif ((x * y) <= 1e+176)
		tmp = t_0;
	else
		tmp = t_1;
	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[(N[(x * y), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]) / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e+274], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e-162], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 4e-141], N[(N[(x / z), $MachinePrecision] / N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+176], t$95$0, t$95$1]]]]]]

Target

Original	77.2%
Target	93.6%
Herbie	97.1%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.9999999999999998e274 or 1e176 < (*.f64 x y)

   1. Initial program 33.4%

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

   2. Simplified 78.0%

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

      [Start] 33.4	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 78.0	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 95.8%

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

   if -4.9999999999999998e274 < (*.f64 x y) < -5.00000000000000014e-162 or 4.0000000000000002e-141 < (*.f64 x y) < 1e176

   1. Initial program 92.5%

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

   2. Simplified 88.0%

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

      [Start] 92.5	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 88.0	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 96.1%

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

   if -5.00000000000000014e-162 < (*.f64 x y) < 4.0000000000000002e-141

   1. Initial program 72.2%

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

   2. Simplified 78.6%

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

      [Start] 72.2	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 78.6	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 99.0%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+274}:\\ \;\;\;\;\frac{-1}{\left(z \cdot \frac{z}{x}\right) \cdot \frac{-1 - z}{y}}\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\frac{x \cdot y}{z + 1}}{z \cdot \left(-z\right)}\\ \mathbf{elif}\;x \cdot y \leq 4 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \mathbf{elif}\;x \cdot y \leq 10^{+176}:\\ \;\;\;\;\frac{-\frac{x \cdot y}{z + 1}}{z \cdot \left(-z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(z \cdot \frac{z}{x}\right) \cdot \frac{-1 - z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	3792

\[\begin{array}{l} t_0 := \frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ t_1 := \frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+228}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	95.2%
Cost	1736

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 3
Accuracy	94.9%
Cost	1736

\[\begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-277}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\frac{x \cdot y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 4
Accuracy	95.6%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \mathbf{if}\;z \leq -86000000000000:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{z}}{z}}{z}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \end{array} \]

Alternative 5
Accuracy	96.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{\frac{\frac{x}{z}}{\frac{z}{y}}}{z}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{z \cdot \left(z \cdot \frac{z + 1}{y}\right)}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	95.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ t_1 := \frac{\frac{x}{z}}{\frac{z}{y}}\\ t_2 := \frac{t_1}{z}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	90.5%
Cost	841

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

Alternative 8
Accuracy	93.0%
Cost	841

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

Alternative 9
Accuracy	93.0%
Cost	841

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

Alternative 10
Accuracy	90.2%
Cost	840

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

Alternative 11
Accuracy	90.4%
Cost	840

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

Alternative 12
Accuracy	93.0%
Cost	840

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

Alternative 13
Accuracy	93.3%
Cost	840

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

Alternative 14
Accuracy	70.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 15
Accuracy	72.7%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 16
Accuracy	73.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 17
Accuracy	64.3%
Cost	448

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

Reproduce

herbie shell --seed 2023141 
(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))))