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

Percentage Accurate: 89.7% → 88.6%

Time: 3.5s

Alternatives: 5

Speedup: N/A×

Specification

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: 89.7% 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: 88.6% 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 90.7%

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

Alternative 2: 75.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y - t\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\left(x - 1\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log y) t)))
   (if (<= t -1.2e+90) t_1 (if (<= t 5e+86) (* (- x 1.0) (log y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) - t;
	double tmp;
	if (t <= -1.2e+90) {
		tmp = t_1;
	} else if (t <= 5e+86) {
		tmp = (x - 1.0) * log(y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = log(y) - t
    if (t <= (-1.2d+90)) then
        tmp = t_1
    else if (t <= 5d+86) then
        tmp = (x - 1.0d0) * log(y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) - t;
	double tmp;
	if (t <= -1.2e+90) {
		tmp = t_1;
	} else if (t <= 5e+86) {
		tmp = (x - 1.0) * Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) - t
	tmp = 0
	if t <= -1.2e+90:
		tmp = t_1
	elif t <= 5e+86:
		tmp = (x - 1.0) * math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) - t)
	tmp = 0.0
	if (t <= -1.2e+90)
		tmp = t_1;
	elseif (t <= 5e+86)
		tmp = Float64(Float64(x - 1.0) * log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) - t;
	tmp = 0.0;
	if (t <= -1.2e+90)
		tmp = t_1;
	elseif (t <= 5e+86)
		tmp = (x - 1.0) * log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t, -1.2e+90], t$95$1, If[LessEqual[t, 5e+86], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.20000000000000005e90 or 4.9999999999999998e86 < t

   1. Initial program 98.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 19.2%

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

   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 19.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(x - 1\right) \cdot \log y \]

   11. Applied rewrites 17.1%

       \[\leadsto \left(\left(x - 1\right) - t\right) \cdot \log y \]

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 80.3%

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

   if -1.20000000000000005e90 < t < 4.9999999999999998e86

   1. Initial program 84.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 78.3%

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

   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 78.3%

      \[\leadsto \left(x - 1\right) \cdot \color{blue}{\log y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 35.2% 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 51.5%

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

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 51.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(x - 1\right) \cdot \log y \]

11. Applied rewrites 50.4%

    \[\leadsto \left(\left(x - 1\right) - t\right) \cdot \log y \]

12. Taylor expanded in y around 0

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

14. Applied rewrites 40.9%

    \[\leadsto \color{blue}{\log y - t} \]
15. Add Preprocessing

Alternative 4: 34.2% 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 40.0%

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

Alternative 5: 5.0% 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 90.7%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 51.5%

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

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 51.5%

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

9. Taylor expanded in x around -inf

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

11. Applied rewrites 4.3%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))