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

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 12

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: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-302}:\\ \;\;\;\;{y}^{y} \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 -1e-302) (* (pow y y) (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 <= -1e-302) {
		tmp = pow(y, y) * 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 <= (-1d-302)) then
        tmp = (y ** y) * 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 <= -1e-302) {
		tmp = Math.pow(y, y) * 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 <= -1e-302:
		tmp = math.pow(y, y) * 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 <= -1e-302)
		tmp = Float64((y ^ y) * 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 <= -1e-302)
		tmp = (y ^ y) * 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, -1e-302], N[(N[Power[y, y], $MachinePrecision] * N[Exp[N[(x - z), $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)) < -9.9999999999999996e-303

   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 -9.9999999999999996e-303 < (*.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 88.8%

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

Alternative 3: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+88}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;e^{y \cdot \log y - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e+88)
   (exp x)
   (if (<= x 1.75e-13) (exp (- (* y (log y)) z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e+88) {
		tmp = exp(x);
	} else if (x <= 1.75e-13) {
		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 <= (-6.2d+88)) then
        tmp = exp(x)
    else if (x <= 1.75d-13) 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 <= -6.2e+88) {
		tmp = Math.exp(x);
	} else if (x <= 1.75e-13) {
		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 <= -6.2e+88:
		tmp = math.exp(x)
	elif x <= 1.75e-13:
		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 <= -6.2e+88)
		tmp = exp(x);
	elseif (x <= 1.75e-13)
		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 <= -6.2e+88)
		tmp = exp(x);
	elseif (x <= 1.75e-13)
		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, -6.2e+88], N[Exp[x], $MachinePrecision], If[LessEqual[x, 1.75e-13], 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 < -6.2000000000000003e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 83.6%

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

   if -6.2000000000000003e88 < x < 1.7500000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 96.9%

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

   if 1.7500000000000001e-13 < 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 97.2%

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

      4. *-commutative 97.2%

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

      5. exp-to-pow 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in z around 0 93.2%

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

      1. *-commutative 93.2%

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

   7. Simplified 93.2%

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

Alternative 4: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -390000 \lor \neg \left(z \leq 2.8 \cdot 10^{+192}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -390000.0) (not (<= z 2.8e+192)))
   (exp (- z))
   (* (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -390000.0) || !(z <= 2.8e+192)) {
		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 <= (-390000.0d0)) .or. (.not. (z <= 2.8d+192))) 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 <= -390000.0) || !(z <= 2.8e+192)) {
		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 <= -390000.0) or not (z <= 2.8e+192):
		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 <= -390000.0) || !(z <= 2.8e+192))
		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 <= -390000.0) || ~((z <= 2.8e+192)))
		tmp = exp(-z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -390000.0], N[Not[LessEqual[z, 2.8e+192]], $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 < -3.9e5 or 2.79999999999999976e192 < 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 88.1%

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

      1. neg-mul-1 88.1%

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

   5. Simplified 88.1%

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

   if -3.9e5 < z < 2.79999999999999976e192

   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 80.6%

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

      4. *-commutative 80.6%

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

      5. exp-to-pow 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in z around 0 83.4%

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

      1. *-commutative 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -390000 \lor \neg \left(z \leq 2.8 \cdot 10^{+192}\right):\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+65}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-84}:\\ \;\;\;\;\frac{{y}^{y}}{e^{z}}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y} \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+65)
   (exp x)
   (if (<= x 1.95e-84) (/ (pow y y) (exp z)) (* (pow y y) (exp x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+65) {
		tmp = exp(x);
	} else if (x <= 1.95e-84) {
		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 <= (-3d+65)) then
        tmp = exp(x)
    else if (x <= 1.95d-84) 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 <= -3e+65) {
		tmp = Math.exp(x);
	} else if (x <= 1.95e-84) {
		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 <= -3e+65:
		tmp = math.exp(x)
	elif x <= 1.95e-84:
		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 <= -3e+65)
		tmp = exp(x);
	elseif (x <= 1.95e-84)
		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 <= -3e+65)
		tmp = exp(x);
	elseif (x <= 1.95e-84)
		tmp = (y ^ y) / exp(z);
	else
		tmp = (y ^ y) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+65], N[Exp[x], $MachinePrecision], If[LessEqual[x, 1.95e-84], 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 < -3.0000000000000002e65

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 79.1%

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

   if -3.0000000000000002e65 < x < 1.95000000000000011e-84

   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 86.1%

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

      4. *-commutative 86.1%

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

      5. exp-to-pow 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. rec-exp 85.6%

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

      2. associate-*l/ 85.6%

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

      3. *-lft-identity 85.6%

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

   7. Simplified 85.6%

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

   if 1.95000000000000011e-84 < 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 94.3%

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

      4. *-commutative 94.3%

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

      5. exp-to-pow 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in z around 0 91.0%

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

      1. *-commutative 91.0%

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

   7. Simplified 91.0%

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

Alternative 6: 73.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-z}\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-92}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;y \leq 0.65:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- z))))
   (if (<= y 2.8e-238)
     t_0
     (if (<= y 7.2e-92) (exp x) (if (<= y 0.65) t_0 (pow y y))))))

C

double code(double x, double y, double z) {
	double t_0 = exp(-z);
	double tmp;
	if (y <= 2.8e-238) {
		tmp = t_0;
	} else if (y <= 7.2e-92) {
		tmp = exp(x);
	} else if (y <= 0.65) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-z)
    if (y <= 2.8d-238) then
        tmp = t_0
    else if (y <= 7.2d-92) then
        tmp = exp(x)
    else if (y <= 0.65d0) then
        tmp = t_0
    else
        tmp = y ** y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp(-z);
	double tmp;
	if (y <= 2.8e-238) {
		tmp = t_0;
	} else if (y <= 7.2e-92) {
		tmp = Math.exp(x);
	} else if (y <= 0.65) {
		tmp = t_0;
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp(-z)
	tmp = 0
	if y <= 2.8e-238:
		tmp = t_0
	elif y <= 7.2e-92:
		tmp = math.exp(x)
	elif y <= 0.65:
		tmp = t_0
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(-z))
	tmp = 0.0
	if (y <= 2.8e-238)
		tmp = t_0;
	elseif (y <= 7.2e-92)
		tmp = exp(x);
	elseif (y <= 0.65)
		tmp = t_0;
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp(-z);
	tmp = 0.0;
	if (y <= 2.8e-238)
		tmp = t_0;
	elseif (y <= 7.2e-92)
		tmp = exp(x);
	elseif (y <= 0.65)
		tmp = t_0;
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[(-z)], $MachinePrecision]}, If[LessEqual[y, 2.8e-238], t$95$0, If[LessEqual[y, 7.2e-92], N[Exp[x], $MachinePrecision], If[LessEqual[y, 0.65], t$95$0, N[Power[y, y], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.80000000000000004e-238 or 7.20000000000000032e-92 < y < 0.650000000000000022

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 75.6%

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

      1. neg-mul-1 75.6%

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

   5. Simplified 75.6%

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

   if 2.80000000000000004e-238 < y < 7.20000000000000032e-92

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 88.6%

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

   if 0.650000000000000022 < 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 65.2%

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

      4. *-commutative 65.2%

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

      5. exp-to-pow 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in z around 0 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in x around 0 83.6%

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

Alternative 7: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+64} \lor \neg \left(x \leq 1.75 \cdot 10^{-13}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e+64) (not (<= x 1.75e-13))) (exp x) (exp (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+64) || !(x <= 1.75e-13)) {
		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 <= (-5d+64)) .or. (.not. (x <= 1.75d-13))) 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 <= -5e+64) || !(x <= 1.75e-13)) {
		tmp = Math.exp(x);
	} else {
		tmp = Math.exp(-z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5e+64) or not (x <= 1.75e-13):
		tmp = math.exp(x)
	else:
		tmp = math.exp(-z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e+64) || !(x <= 1.75e-13))
		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 <= -5e+64) || ~((x <= 1.75e-13)))
		tmp = exp(x);
	else
		tmp = exp(-z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e+64], N[Not[LessEqual[x, 1.75e-13]], $MachinePrecision]], N[Exp[x], $MachinePrecision], N[Exp[(-z)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e64 or 1.7500000000000001e-13 < 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 86.8%

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

   if -5e64 < x < 1.7500000000000001e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 66.3%

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

      1. neg-mul-1 66.3%

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

   5. Simplified 66.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+64} \lor \neg \left(x \leq 1.75 \cdot 10^{-13}\right):\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% 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 55.8%

   \[\leadsto e^{\color{blue}{x}} \]
4. Add Preprocessing

Alternative 9: 31.6% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0138:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0138)
   (+ 1.0 (* x (+ 1.0 (* x 0.5))))
   (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0138) {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	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 <= (-0.0138d0)) then
        tmp = 1.0d0 + (x * (1.0d0 + (x * 0.5d0)))
    else
        tmp = 1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0138) {
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.0138:
		tmp = 1.0 + (x * (1.0 + (x * 0.5)))
	else:
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0138)
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.0138)
		tmp = 1.0 + (x * (1.0 + (x * 0.5)));
	else
		tmp = 1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0138], N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0138

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 66.2%

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

   4. Taylor expanded in x around 0 10.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + 0.5 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 10.5%

         \[\leadsto 1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

   6. Simplified 10.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]

   if -0.0138 < 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 52.5%

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

   4. Taylor expanded in x around 0 44.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 44.2%

         \[\leadsto 1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right) \]

   6. Simplified 44.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 28.2% accurate, 23.0× speedup

Math

\[\begin{array}{l} \\ 1 + x \cdot \left(1 + x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 1.0 (* x (+ 1.0 (* x 0.5)))))

C

double code(double x, double y, double z) {
	return 1.0 + (x * (1.0 + (x * 0.5)));
}

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 + (x * (1.0d0 + (x * 0.5d0)))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + (x * (1.0 + (x * 0.5)));
}

Python

def code(x, y, z):
	return 1.0 + (x * (1.0 + (x * 0.5)))

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + (x * (1.0 + (x * 0.5)));
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 55.8%

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

4. Taylor expanded in x around 0 32.9%

   \[\leadsto \color{blue}{1 + x \cdot \left(1 + 0.5 \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 32.9%

      \[\leadsto 1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right) \]

6. Simplified 32.9%

   \[\leadsto \color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)} \]
7. Add Preprocessing

Alternative 11: 14.7% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x + 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(x + 1.0), $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 55.8%

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

4. Taylor expanded in x around 0 16.0%

   \[\leadsto \color{blue}{1 + x} \]
5. Step-by-step derivation

   1. +-commutative 16.0%

      \[\leadsto \color{blue}{x + 1} \]

6. Simplified 16.0%

   \[\leadsto \color{blue}{x + 1} \]
7. Add Preprocessing

Alternative 12: 14.5% accurate, 207.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

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 55.8%

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

4. Taylor expanded in x around 0 15.6%

   \[\leadsto \color{blue}{1} \]
5. 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 2024111 
(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)))