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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 6

Speedup: 1.0×

• 58.1% 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. Add Preprocessing

3. Final simplification 100.0%

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

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 100000000000:\\ \;\;\;\;\frac{{y}^{y}}{e^{z - x}}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 100000000000.0) (/ (pow y y) (exp (- z x))) (exp (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 100000000000.0) {
		tmp = pow(y, y) / exp((z - x));
	} 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 <= 100000000000.0d0) then
        tmp = (y ** y) / exp((z - x))
    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 <= 100000000000.0) {
		tmp = Math.pow(y, y) / Math.exp((z - x));
	} 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 <= 100000000000.0:
		tmp = math.pow(y, y) / math.exp((z - x))
	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 <= 100000000000.0)
		tmp = Float64((y ^ y) / exp(Float64(z - x)));
	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 <= 100000000000.0)
		tmp = (y ^ y) / exp((z - x));
	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, 100000000000.0], N[(N[Power[y, y], $MachinePrecision] / N[Exp[N[(z - x), $MachinePrecision]], $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)) < 1e11

   1. Initial program 100.0%

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

      1. associate--l+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. exp-diff 100.0%

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

      5. *-commutative 100.0%

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

      6. exp-to-pow 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{y}^{y}}{e^{z - x}}} \]
   4. Add Preprocessing

   if 1e11 < (*.f64 y (log.f64 y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.6%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 100000000000:\\ \;\;\;\;\frac{{y}^{y}}{e^{z - x}}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e-13) (exp (- x z)) (exp (- (* y (log y)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-13) {
		tmp = exp((x - z));
	} else {
		tmp = exp(((y * log(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
    real(8) :: tmp
    if (y <= 8.5d-13) then
        tmp = exp((x - z))
    else
        tmp = exp(((y * log(y)) - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e-13) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(((y * Math.log(y)) - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8.5e-13:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(((y * math.log(y)) - z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e-13)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(Float64(y * log(y)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8.5e-13)
		tmp = exp((x - z));
	else
		tmp = exp(((y * log(y)) - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e-13], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 8.5000000000000001e-13 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 93.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+17} \lor \neg \left(y \leq 5.2 \cdot 10^{+69}\right) \land y \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 8.5e+17) (and (not (<= y 5.2e+69)) (<= y 2.4e+133)))
   (exp (- x z))
   (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 8.5e+17) || (!(y <= 5.2e+69) && (y <= 2.4e+133))) {
		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 <= 8.5d+17) .or. (.not. (y <= 5.2d+69)) .and. (y <= 2.4d+133)) 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 <= 8.5e+17) || (!(y <= 5.2e+69) && (y <= 2.4e+133))) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 8.5e+17) or (not (y <= 5.2e+69) and (y <= 2.4e+133)):
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 8.5e+17) || (!(y <= 5.2e+69) && (y <= 2.4e+133)))
		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 <= 8.5e+17) || (~((y <= 5.2e+69)) && (y <= 2.4e+133)))
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 8.5e+17], And[N[Not[LessEqual[y, 5.2e+69]], $MachinePrecision], LessEqual[y, 2.4e+133]]], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5e17 or 5.2000000000000004e69 < y < 2.3999999999999999e133

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.1%

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

   if 8.5e17 < y < 5.2000000000000004e69 or 2.3999999999999999e133 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 94.7%

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

   4. Taylor expanded in z around 0 84.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+17} \lor \neg \left(y \leq 5.2 \cdot 10^{+69}\right) \land y \leq 2.4 \cdot 10^{+133}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-5} \lor \neg \left(z \leq 7200000\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.6e-5) (not (<= z 7200000.0))) (exp (- z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.6e-5) || !(z <= 7200000.0)) {
		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 ((z <= (-1.6d-5)) .or. (.not. (z <= 7200000.0d0))) 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 ((z <= -1.6e-5) || !(z <= 7200000.0)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.6e-5) or not (z <= 7200000.0):
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.6e-5) || !(z <= 7200000.0))
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.6e-5) || ~((z <= 7200000.0)))
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.6e-5], N[Not[LessEqual[z, 7200000.0]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.59999999999999993e-5 or 7.2e6 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 80.8%

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

      1. neg-mul-1 80.8%

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

   5. Simplified 80.8%

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

   if -1.59999999999999993e-5 < z < 7.2e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 68.6%

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

   4. Taylor expanded in z around 0 68.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-5} \lor \neg \left(z \leq 7200000\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.4% 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. Add Preprocessing

3. Taylor expanded in z around inf 53.2%

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

   1. neg-mul-1 53.2%

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

5. Simplified 53.2%

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

6. Final simplification 53.2%

   \[\leadsto e^{-z} \]
7. Add Preprocessing

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 2024048 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

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

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