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

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 8

Speedup: 1.0×

• 58.4% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 200000:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{elif}\;t\_0 \leq 10^{+47}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 200000.0)
     (* (pow y y) (exp (- x z)))
     (if (<= t_0 1e+47) (exp 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 <= 200000.0) {
		tmp = pow(y, y) * exp((x - z));
	} else if (t_0 <= 1e+47) {
		tmp = exp(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 <= 200000.0d0) then
        tmp = (y ** y) * exp((x - z))
    else if (t_0 <= 1d+47) then
        tmp = exp(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 <= 200000.0) {
		tmp = Math.pow(y, y) * Math.exp((x - z));
	} else if (t_0 <= 1e+47) {
		tmp = Math.exp(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 <= 200000.0:
		tmp = math.pow(y, y) * math.exp((x - z))
	elif t_0 <= 1e+47:
		tmp = math.exp(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 <= 200000.0)
		tmp = Float64((y ^ y) * exp(Float64(x - z)));
	elseif (t_0 <= 1e+47)
		tmp = exp(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 <= 200000.0)
		tmp = (y ^ y) * exp((x - z));
	elseif (t_0 <= 1e+47)
		tmp = exp(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, 200000.0], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+47], N[Exp[x], $MachinePrecision], N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < 2e5

   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 100.0%

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

      4. *-commutative 100.0%

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

      5. exp-to-pow 100.0%

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

   3. Simplified 100.0%

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

   if 2e5 < (*.f64 y (log.f64 y)) < 1e47

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 86.2%

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

   if 1e47 < (*.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 92.2%

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

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{+145}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 0.013:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.95e+145)
   (exp x)
   (if (<= x 0.013) (exp (- (* y (log y)) z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e+145) {
		tmp = exp(x);
	} else if (x <= 0.013) {
		tmp = exp(((y * log(y)) - z));
	} else {
		tmp = pow(y, y) * exp(x);
	}
	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 (x <= (-1.95d+145)) then
        tmp = exp(x)
    else if (x <= 0.013d0) then
        tmp = exp(((y * log(y)) - z))
    else
        tmp = (y ** y) * exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.95e+145) {
		tmp = Math.exp(x);
	} else if (x <= 0.013) {
		tmp = Math.exp(((y * Math.log(y)) - z));
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.95e+145:
		tmp = math.exp(x)
	elif x <= 0.013:
		tmp = math.exp(((y * math.log(y)) - z))
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.95e+145)
		tmp = exp(x);
	elseif (x <= 0.013)
		tmp = exp(Float64(Float64(y * log(y)) - z));
	else
		tmp = Float64((y ^ y) * exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.95e+145)
		tmp = exp(x);
	elseif (x <= 0.013)
		tmp = exp(((y * log(y)) - z));
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.95e+145], N[Exp[x], $MachinePrecision], If[LessEqual[x, 0.013], N[Exp[N[(N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9499999999999999e145

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.6%

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

   if -1.9499999999999999e145 < x < 0.0129999999999999994

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.7%

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

   if 0.0129999999999999994 < x

   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 94.5%

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

      4. *-commutative 94.5%

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

      5. exp-to-pow 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in z around 0 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -680 \lor \neg \left(z \leq 3.3 \cdot 10^{+81}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -680.0) (not (<= z 3.3e+81)))
   (exp (- z))
   (* (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -680.0) || !(z <= 3.3e+81)) {
		tmp = exp(-z);
	} else {
		tmp = pow(y, y) * exp(x);
	}
	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 <= (-680.0d0)) .or. (.not. (z <= 3.3d+81))) then
        tmp = exp(-z)
    else
        tmp = (y ** y) * exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -680.0) || !(z <= 3.3e+81)) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -680.0) or not (z <= 3.3e+81):
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -680.0) || !(z <= 3.3e+81))
		tmp = exp(Float64(-z));
	else
		tmp = Float64((y ^ y) * exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -680.0) || ~((z <= 3.3e+81)))
		tmp = exp(-z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -680.0], N[Not[LessEqual[z, 3.3e+81]], $MachinePrecision]], N[Exp[(-z)], $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -680 or 3.3e81 < 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 81.5%

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

      1. neg-mul-1 81.5%

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

   5. Simplified 81.5%

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

   if -680 < z < 3.3e81

   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 83.3%

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

      4. *-commutative 83.3%

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

      5. exp-to-pow 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in z around 0 83.8%

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

      1. *-commutative 83.8%

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

   7. Simplified 83.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -680 \lor \neg \left(z \leq 3.3 \cdot 10^{+81}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+145}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 0.0024:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.7e+145)
   (exp x)
   (if (<= x 0.0024) (/ (pow y y) (exp z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+145) {
		tmp = exp(x);
	} else if (x <= 0.0024) {
		tmp = pow(y, y) / exp(z);
	} else {
		tmp = pow(y, y) * exp(x);
	}
	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 (x <= (-4.7d+145)) then
        tmp = exp(x)
    else if (x <= 0.0024d0) then
        tmp = (y ** y) / exp(z)
    else
        tmp = (y ** y) * exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.7e+145) {
		tmp = Math.exp(x);
	} else if (x <= 0.0024) {
		tmp = Math.pow(y, y) / Math.exp(z);
	} else {
		tmp = Math.pow(y, y) * Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.7e+145:
		tmp = math.exp(x)
	elif x <= 0.0024:
		tmp = math.pow(y, y) / math.exp(z)
	else:
		tmp = math.pow(y, y) * math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.7e+145)
		tmp = exp(x);
	elseif (x <= 0.0024)
		tmp = Float64((y ^ y) / exp(z));
	else
		tmp = Float64((y ^ y) * exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4.7e+145)
		tmp = exp(x);
	elseif (x <= 0.0024)
		tmp = (y ^ y) / exp(z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.7e+145], N[Exp[x], $MachinePrecision], If[LessEqual[x, 0.0024], N[(N[Power[y, y], $MachinePrecision] / N[Exp[z], $MachinePrecision]), $MachinePrecision], N[(N[Power[y, y], $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7000000000000002e145

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 84.6%

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

   if -4.7000000000000002e145 < x < 0.00239999999999999979

   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 82.8%

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

      4. *-commutative 82.8%

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

      5. exp-to-pow 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in x around 0 84.3%

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

      1. *-commutative 84.3%

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

      2. exp-to-pow 84.3%

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

      3. *-commutative 84.3%

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

      4. exp-sum 93.7%

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

      5. sub-neg 93.7%

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

      6. exp-diff 84.3%

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

      7. *-commutative 84.3%

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

      8. exp-to-pow 84.3%

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

   7. Simplified 84.3%

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

   if 0.00239999999999999979 < x

   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 94.5%

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

      4. *-commutative 94.5%

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

      5. exp-to-pow 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in z around 0 87.5%

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

      1. *-commutative 87.5%

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

   7. Simplified 87.5%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+44} \lor \neg \left(x \leq 0.023\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.8e+44) (not (<= x 0.023))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e+44) || !(x <= 0.023)) {
		tmp = exp(x);
	} else {
		tmp = exp(-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 ((x <= (-3.8d+44)) .or. (.not. (x <= 0.023d0))) then
        tmp = exp(x)
    else
        tmp = exp(-z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e+44) || !(x <= 0.023)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.8e+44) or not (x <= 0.023):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.8e+44) || !(x <= 0.023))
		tmp = exp(x);
	else
		tmp = exp(Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.8e+44) || ~((x <= 0.023)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.8e+44], N[Not[LessEqual[x, 0.023]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e44 or 0.023 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 78.2%

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

   if -3.8000000000000002e44 < x < 0.023

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 67.1%

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

      1. neg-mul-1 67.1%

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

   5. Simplified 67.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+44} \lor \neg \left(x \leq 0.023\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-14}:\\ \;\;\;\;e^{-z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+45}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.5e-14) (exp (- z)) (if (<= y 3e+45) (exp x) (pow y y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.5e-14) {
		tmp = exp(-z);
	} else if (y <= 3e+45) {
		tmp = exp(x);
	} 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 <= 1.5d-14) then
        tmp = exp(-z)
    else if (y <= 3d+45) then
        tmp = exp(x)
    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 <= 1.5e-14) {
		tmp = Math.exp(-z);
	} else if (y <= 3e+45) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.5e-14:
		tmp = math.exp(-z)
	elif y <= 3e+45:
		tmp = math.exp(x)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.5e-14)
		tmp = exp(Float64(-z));
	elseif (y <= 3e+45)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.5e-14)
		tmp = exp(-z);
	elseif (y <= 3e+45)
		tmp = exp(x);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.5e-14], N[Exp[(-z)], $MachinePrecision], If[LessEqual[y, 3e+45], N[Exp[x], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4999999999999999e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 76.3%

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

      1. neg-mul-1 76.3%

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

   5. Simplified 76.3%

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

   if 1.4999999999999999e-14 < y < 3.00000000000000011e45

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 70.7%

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

   if 3.00000000000000011e45 < 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 64.0%

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

      4. *-commutative 64.0%

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

      5. exp-to-pow 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in z around 0 71.1%

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

      1. *-commutative 71.1%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in x around 0 84.3%

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

Alternative 8: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e^{x} \end{array} \]

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[Exp[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf 50.2%

   \[\leadsto e^{\color{blue}{x}} \]
4. 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 2024103 
(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)))