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

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

Alternatives: 5

Speedup: 1.0×

• 52.1% 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. Final simplification 100.0%

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

Alternative 2: 90.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+119}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= y 5e+119) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e+119) {
		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 <= 5d+119) 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 <= 5e+119) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e+119:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e+119)
		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 <= 5e+119)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e+119], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e119

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 89.3%

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

   if 4.9999999999999999e119 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 93.5%

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

      1. log-pow 88.3%

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

      2. div-exp 73.8%

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

      3. rem-exp-log 73.8%

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

   4. Simplified 73.8%

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

   5. Taylor expanded in z around 0 88.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+119}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 3: 73.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 35000000000000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 35000000000000.0) (exp (- z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 35000000000000.0) {
		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 <= 35000000000000.0d0) 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 <= 35000000000000.0) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 35000000000000.0:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 35000000000000.0)
		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 <= 35000000000000.0)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 35000000000000.0], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e13

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 72.0%

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

      1. log-pow 70.3%

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

      2. div-exp 70.3%

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

      3. rem-exp-log 70.3%

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

   4. Simplified 70.3%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. rec-exp 70.5%

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

   7. Simplified 70.5%

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

   if 3.5e13 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.6%

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

      1. log-pow 80.0%

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

      2. div-exp 69.3%

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

      3. rem-exp-log 69.3%

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

   4. Simplified 69.3%

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

   5. Taylor expanded in z around 0 80.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 35000000000000:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]

Alternative 4: 51.3% 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 x around 0 81.1%

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

   1. log-pow 75.4%

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

   2. div-exp 69.8%

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

   3. rem-exp-log 69.8%

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

4. Simplified 69.8%

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

5. Taylor expanded in y around 0 51.3%

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

   1. rec-exp 51.3%

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

7. Simplified 51.3%

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

8. Final simplification 51.3%

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

Alternative 5: 14.6% accurate, 69.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 - 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 = 1.0d0 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 81.1%

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

   1. log-pow 75.4%

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

   2. div-exp 69.8%

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

   3. rem-exp-log 69.8%

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

4. Simplified 69.8%

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

5. Taylor expanded in z around 0 35.6%

   \[\leadsto \color{blue}{-1 \cdot \left({y}^{y} \cdot z\right) + {y}^{y}} \]
6. Step-by-step derivation

   1. +-commutative 35.6%

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

   2. mul-1-neg 35.6%

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

   3. unsub-neg 35.6%

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

   4. *-rgt-identity 35.6%

      \[\leadsto \color{blue}{{y}^{y} \cdot 1} - {y}^{y} \cdot z \]

   5. distribute-lft-out-- 48.7%

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

7. Simplified 48.7%

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

8. Taylor expanded in y around 0 13.3%

   \[\leadsto \color{blue}{1 - z} \]

9. Final simplification 13.3%

   \[\leadsto 1 - z \]

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 2023238 
(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)))