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

Percentage Accurate: 89.1% → 89.1%

Time: 3.5s

Alternatives: 8

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

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

Initial Program: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

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

Alternative 1: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

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

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{+79}:\\ \;\;\;\;\log y - t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+68}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.3e+79)
   (- (log y) t)
   (if (<= t 8e+68) (* (- x 1.0) (log y)) (- (log (- 1.0 y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e+79) {
		tmp = log(y) - t;
	} else if (t <= 8e+68) {
		tmp = (x - 1.0) * log(y);
	} else {
		tmp = log((1.0 - y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.3d+79)) then
        tmp = log(y) - t
    else if (t <= 8d+68) then
        tmp = (x - 1.0d0) * log(y)
    else
        tmp = log((1.0d0 - y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.3e+79) {
		tmp = Math.log(y) - t;
	} else if (t <= 8e+68) {
		tmp = (x - 1.0) * Math.log(y);
	} else {
		tmp = Math.log((1.0 - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.3e+79:
		tmp = math.log(y) - t
	elif t <= 8e+68:
		tmp = (x - 1.0) * math.log(y)
	else:
		tmp = math.log((1.0 - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.3e+79)
		tmp = Float64(log(y) - t);
	elseif (t <= 8e+68)
		tmp = Float64(Float64(x - 1.0) * log(y));
	else
		tmp = Float64(log(Float64(1.0 - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.3e+79)
		tmp = log(y) - t;
	elseif (t <= 8e+68)
		tmp = (x - 1.0) * log(y);
	else
		tmp = log((1.0 - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.3e+79], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 8e+68], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.3e79

   1. Initial program 95.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.1%

      \[\leadsto \color{blue}{\left(x - 1\right)} - t \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.5%

      \[\leadsto \log y - t \]

   if -2.3e79 < t < 7.99999999999999962e68

   1. Initial program 82.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.4%

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

   if 7.99999999999999962e68 < t

   1. Initial program 100.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.8%

      \[\leadsto \log \left(1 - y\right) - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(x - 1\right) \cdot \log y - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* (- x 1.0) (log y)) t))

C

double code(double x, double y, double z, double t) {
	return ((x - 1.0) * log(y)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x - 1.0d0) * log(y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x - 1.0) * Math.log(y)) - t;
}

Python

def code(x, y, z, t):
	return ((x - 1.0) * math.log(y)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x - 1.0) * log(y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x - 1.0) * log(y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 87.5%

   \[\leadsto \color{blue}{\left(x - 1\right) \cdot \log y} - t \]
6. Add Preprocessing

Alternative 4: 35.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \log \left(1 - y\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (log (- 1.0 y)) t))

C

double code(double x, double y, double z, double t) {
	return log((1.0 - y)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = log((1.0d0 - y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.log((1.0 - y)) - t;
}

Python

def code(x, y, z, t):
	return math.log((1.0 - y)) - t

Julia

function code(x, y, z, t)
	return Float64(log(Float64(1.0 - y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = log((1.0 - y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 87.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 35.2%

   \[\leadsto \log \left(1 - y\right) - t \]
9. Add Preprocessing

Alternative 5: 34.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \log y - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (log y) t))

C

double code(double x, double y, double z, double t) {
	return log(y) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = log(y) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.log(y) - t;
}

Python

def code(x, y, z, t):
	return math.log(y) - t

Julia

function code(x, y, z, t)
	return Float64(log(y) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = log(y) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\left(x - 1\right)} - t \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 35.0%

   \[\leadsto \log y - t \]
9. Add Preprocessing

Alternative 6: 34.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \left(x - 1\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (- x 1.0) t))

C

double code(double x, double y, double z, double t) {
	return (x - 1.0) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - 1.0d0) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - 1.0) - t;
}

Python

def code(x, y, z, t):
	return (x - 1.0) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(x - 1.0) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - 1.0) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - 1.0), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\left(x - 1\right)} - t \]
6. Add Preprocessing

Alternative 7: 5.1% accurate, 56.5× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 1.0 y))

C

double code(double x, double y, double z, double t) {
	return 1.0 - y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 1.0d0 - y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - y;
}

Python

def code(x, y, z, t):
	return 1.0 - y

Julia

function code(x, y, z, t)
	return Float64(1.0 - y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 1.0 - y;
end

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - y), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 87.5%

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

6. Taylor expanded in x around 0

   \[\leadsto -1 \cdot \color{blue}{\log y} - t \]

8. Applied rewrites 59.8%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 5.0%

    \[\leadsto \color{blue}{1 - y} \]
12. Add Preprocessing

Alternative 8: 2.0% accurate, 56.5× speedup

Math

\[\begin{array}{l} \\ x - 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x 1.0))

C

double code(double x, double y, double z, double t) {
	return x - 1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - 1.0;
}

Python

def code(x, y, z, t):
	return x - 1.0

Julia

function code(x, y, z, t)
	return Float64(x - 1.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - 1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(x - 1.0), $MachinePrecision]

Derivation

1. Initial program 88.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 2.0%

   \[\leadsto x - \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  :pre (TRUE)
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))