Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

Alternatives: 5

Speedup: 1.0×

• 52.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]

2. Final simplification 100.0%

   \[\leadsto e^{\left(x + y \cdot \log y\right) - z} \]

Alternative 2: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+81}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 5e+81) (exp (- x z)) (exp (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 5e+81) {
		tmp = exp((x - z));
	} else {
		tmp = exp((t_0 - 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (t_0 <= 5d+81) then
        tmp = exp((x - z))
    else
        tmp = exp((t_0 - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (t_0 <= 5e+81) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp((t_0 - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if t_0 <= 5e+81:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp((t_0 - z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= 5e+81)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(t_0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (t_0 <= 5e+81)
		tmp = exp((x - z));
	else
		tmp = exp((t_0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e+81], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 4.9999999999999998e81

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around inf 97.6%

      \[\leadsto e^{\color{blue}{x} - z} \]

   if 4.9999999999999998e81 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around 0 94.9%

      \[\leadsto e^{\color{blue}{y \cdot \log y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 5 \cdot 10^{+81}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]

Alternative 3: 89.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+144) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+144) {
		tmp = exp((x - z));
	} else {
		tmp = pow(y, y);
	}
	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
    real(8) :: tmp
    if (y <= 2.4d+144) then
        tmp = exp((x - z))
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+144) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.4e+144:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+144)
		tmp = exp(Float64(x - z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.4e+144)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e+144], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e144

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in x around inf 94.3%

      \[\leadsto e^{\color{blue}{x} - z} \]

   if 2.4000000000000001e144 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 57.1%

         \[\leadsto \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]

      4. *-commutative 57.1%

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \cdot e^{x - z} \]

      5. exp-to-pow 57.1%

         \[\leadsto \color{blue}{{y}^{y}} \cdot e^{x - z} \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]

   4. Taylor expanded in x around 0 74.4%

      \[\leadsto \color{blue}{e^{-z} \cdot {y}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 74.4%

         \[\leadsto \color{blue}{{y}^{y} \cdot e^{-z}} \]

      2. exp-to-pow 74.4%

         \[\leadsto \color{blue}{e^{\log y \cdot y}} \cdot e^{-z} \]

      3. *-commutative 74.4%

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \cdot e^{-z} \]

      4. exp-sum 95.8%

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

      5. sub-neg 95.8%

         \[\leadsto e^{\color{blue}{y \cdot \log y - z}} \]

      6. exp-diff 74.4%

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log y}}{e^{z}}} \]

      7. *-commutative 74.4%

         \[\leadsto \frac{e^{\color{blue}{\log y \cdot y}}}{e^{z}} \]

      8. exp-to-pow 74.4%

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

   6. Simplified 74.4%

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

   7. Taylor expanded in z around 0 91.6%

      \[\leadsto \color{blue}{{y}^{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+144}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 4: 73.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 300000000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 300000000.0) (exp (- z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 300000000.0) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y);
	}
	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
    real(8) :: tmp
    if (y <= 300000000.0d0) then
        tmp = exp(-z)
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 300000000.0) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 300000000.0:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 300000000.0)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 300000000.0)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 300000000.0], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e8

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]

   2. Taylor expanded in z around inf 72.9%

      \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
   3. Step-by-step derivation

      1. neg-mul-1 72.9%

         \[\leadsto e^{\color{blue}{-z}} \]

   4. Simplified 72.9%

      \[\leadsto e^{\color{blue}{-z}} \]

   if 3e8 < y

   1. Initial program 100.0%

      \[e^{\left(x + y \cdot \log y\right) - z} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. associate--l+ 100.0%

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

      3. exp-sum 58.7%

         \[\leadsto \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]

      4. *-commutative 58.7%

         \[\leadsto e^{\color{blue}{\log y \cdot y}} \cdot e^{x - z} \]

      5. exp-to-pow 58.7%

         \[\leadsto \color{blue}{{y}^{y}} \cdot e^{x - z} \]

   3. Simplified 58.7%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]

   4. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{e^{-z} \cdot {y}^{y}} \]
   5. Step-by-step derivation

      1. *-commutative 66.2%

         \[\leadsto \color{blue}{{y}^{y} \cdot e^{-z}} \]

      2. exp-to-pow 66.2%

         \[\leadsto \color{blue}{e^{\log y \cdot y}} \cdot e^{-z} \]

      3. *-commutative 66.2%

         \[\leadsto e^{\color{blue}{y \cdot \log y}} \cdot e^{-z} \]

      4. exp-sum 89.4%

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

      5. sub-neg 89.4%

         \[\leadsto e^{\color{blue}{y \cdot \log y - z}} \]

      6. exp-diff 66.2%

         \[\leadsto \color{blue}{\frac{e^{y \cdot \log y}}{e^{z}}} \]

      7. *-commutative 66.2%

         \[\leadsto \frac{e^{\color{blue}{\log y \cdot y}}}{e^{z}} \]

      8. exp-to-pow 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in z around 0 80.5%

      \[\leadsto \color{blue}{{y}^{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 300000000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 5: 52.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{-z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (exp (- z)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Exp[(-z)], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[e^{\left(x + y \cdot \log y\right) - z} \]

2. Taylor expanded in z around inf 57.2%

   \[\leadsto e^{\color{blue}{-1 \cdot z}} \]
3. Step-by-step derivation

   1. neg-mul-1 57.2%

      \[\leadsto e^{\color{blue}{-z}} \]

4. Simplified 57.2%

   \[\leadsto e^{\color{blue}{-z}} \]

5. Final simplification 57.2%

   \[\leadsto e^{-z} \]

Developer target: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (exp (+ (- x z) (* (log y) y)))

  (exp (- (+ x (* y (log y))) z)))