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

Percentage Accurate: 100.0% → 100.0%

Time: 6.2s

Alternatives: 11

Speedup: 1.0×

• 59.0% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 2: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y + x\\ \mathbf{if}\;t\_0 \leq -40:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+23}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (log y) y) x)))
   (if (<= t_0 -40.0) (exp x) (if (<= t_0 5e+23) (exp (- z)) (pow y y)))))

C

double code(double x, double y, double z) {
	double t_0 = (log(y) * y) + x;
	double tmp;
	if (t_0 <= -40.0) {
		tmp = exp(x);
	} else if (t_0 <= 5e+23) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (log(y) * y) + x
    if (t_0 <= (-40.0d0)) then
        tmp = exp(x)
    else if (t_0 <= 5d+23) 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 t_0 = (Math.log(y) * y) + x;
	double tmp;
	if (t_0 <= -40.0) {
		tmp = Math.exp(x);
	} else if (t_0 <= 5e+23) {
		tmp = Math.exp(-z);
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.log(y) * y) + x
	tmp = 0
	if t_0 <= -40.0:
		tmp = math.exp(x)
	elif t_0 <= 5e+23:
		tmp = math.exp(-z)
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(log(y) * y) + x)
	tmp = 0.0
	if (t_0 <= -40.0)
		tmp = exp(x);
	elseif (t_0 <= 5e+23)
		tmp = exp(Float64(-z));
	else
		tmp = y ^ y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (log(y) * y) + x;
	tmp = 0.0;
	if (t_0 <= -40.0)
		tmp = exp(x);
	elseif (t_0 <= 5e+23)
		tmp = exp(-z);
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$0, -40.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+23], N[Exp[(-z)], $MachinePrecision], N[Power[y, y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 y (log.f64 y))) < -40

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto {y}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.9%

      \[\leadsto {y}^{\color{blue}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 87.9%

       \[\leadsto e^{x} \]

   if -40 < (+.f64 x (*.f64 y (log.f64 y))) < 4.9999999999999999e23

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      2. lower-neg.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 4.9999999999999999e23 < (+.f64 x (*.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 z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto {y}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.8%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y + x \leq -40:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\log y \cdot y + x \leq 5 \cdot 10^{+23}:\\ \;\;\;\;e^{-z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y\\ \mathbf{if}\;t\_0 \leq 50000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) y)))
   (if (<= t_0 50000000.0) (exp (- x z)) (exp (- t_0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * y;
	double tmp;
	if (t_0 <= 50000000.0) {
		tmp = exp((x - z));
	} else {
		tmp = exp((t_0 - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(y) * y
    if (t_0 <= 50000000.0d0) then
        tmp = exp((x - z))
    else
        tmp = exp((t_0 - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) * y;
	double tmp;
	if (t_0 <= 50000000.0) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp((t_0 - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * y
	tmp = 0
	if t_0 <= 50000000.0:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp((t_0 - z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * y)
	tmp = 0.0
	if (t_0 <= 50000000.0)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(t_0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * y;
	tmp = 0.0;
	if (t_0 <= 50000000.0)
		tmp = exp((x - z));
	else
		tmp = exp((t_0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 50000000.0], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - z), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 5e7 < (*.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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 92.6

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

   5. Applied rewrites 92.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 50000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x - z}\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+112}:\\ \;\;\;\;e^{x} \cdot {y}^{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- x z))))
   (if (<= z -1.5e-35) t_0 (if (<= z 6.8e+112) (* (exp x) (pow y y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = exp((x - z));
	double tmp;
	if (z <= -1.5e-35) {
		tmp = t_0;
	} else if (z <= 6.8e+112) {
		tmp = exp(x) * pow(y, y);
	} else {
		tmp = t_0;
	}
	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((x - z))
    if (z <= (-1.5d-35)) then
        tmp = t_0
    else if (z <= 6.8d+112) then
        tmp = exp(x) * (y ** y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.exp((x - z));
	double tmp;
	if (z <= -1.5e-35) {
		tmp = t_0;
	} else if (z <= 6.8e+112) {
		tmp = Math.exp(x) * Math.pow(y, y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.exp((x - z))
	tmp = 0
	if z <= -1.5e-35:
		tmp = t_0
	elif z <= 6.8e+112:
		tmp = math.exp(x) * math.pow(y, y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = exp(Float64(x - z))
	tmp = 0.0
	if (z <= -1.5e-35)
		tmp = t_0;
	elseif (z <= 6.8e+112)
		tmp = Float64(exp(x) * (y ^ y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = exp((x - z));
	tmp = 0.0;
	if (z <= -1.5e-35)
		tmp = t_0;
	elseif (z <= 6.8e+112)
		tmp = exp(x) * (y ^ y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, -1.5e-35], t$95$0, If[LessEqual[z, 6.8e+112], N[(N[Exp[x], $MachinePrecision] * N[Power[y, y], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999994e-35 or 6.79999999999999987e112 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

   if -1.49999999999999994e-35 < z < 6.79999999999999987e112

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-35}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+112}:\\ \;\;\;\;e^{x} \cdot {y}^{y}\\ \mathbf{else}:\\ \;\;\;\;e^{x - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 200000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) 200000000.0) (exp (- x z)) (pow y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= 200000000.0) {
		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 ((log(y) * y) <= 200000000.0d0) 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 ((Math.log(y) * y) <= 200000000.0) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.pow(y, y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (math.log(y) * y) <= 200000000.0:
		tmp = math.exp((x - z))
	else:
		tmp = math.pow(y, y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= 200000000.0)
		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 ((log(y) * y) <= 200000000.0)
		tmp = exp((x - z));
	else
		tmp = y ^ y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], 200000000.0], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Power[y, y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   if 2e8 < (*.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 z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 74.1

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto {y}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 200000000:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;{y}^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;{y}^{y}\\ \mathbf{else}:\\ \;\;\;\;e^{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -45.0) (exp x) (if (<= x 700.0) (pow y y) (exp x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -45.0) {
		tmp = exp(x);
	} else if (x <= 700.0) {
		tmp = pow(y, y);
	} else {
		tmp = 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 <= (-45.0d0)) then
        tmp = exp(x)
    else if (x <= 700.0d0) then
        tmp = y ** y
    else
        tmp = exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -45.0) {
		tmp = Math.exp(x);
	} else if (x <= 700.0) {
		tmp = Math.pow(y, y);
	} else {
		tmp = Math.exp(x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -45.0:
		tmp = math.exp(x)
	elif x <= 700.0:
		tmp = math.pow(y, y)
	else:
		tmp = math.exp(x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -45.0)
		tmp = exp(x);
	elseif (x <= 700.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -45.0)
		tmp = exp(x);
	elseif (x <= 700.0)
		tmp = y ^ y;
	else
		tmp = exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -45.0], N[Exp[x], $MachinePrecision], If[LessEqual[x, 700.0], N[Power[y, y], $MachinePrecision], N[Exp[x], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -45 or 700 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto {y}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.9%

      \[\leadsto {y}^{\color{blue}{y}} \]

   9. Taylor expanded in y around 0

      \[\leadsto e^{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.9%

       \[\leadsto e^{x} \]

   if -45 < x < 700

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. exp-sum N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. exp-to-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-exp.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto {y}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.3%

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

Alternative 7: 52.8% accurate, 2.1× 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 z around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 68.9

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in x around 0

   \[\leadsto {y}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto {y}^{\color{blue}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto e^{x} \]
10. Step-by-step derivation

11. Applied rewrites 51.3%

    \[\leadsto e^{x} \]
12. Add Preprocessing

Alternative 8: 29.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))

C

double code(double x, double y, double z) {
	return fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
}

Julia

function code(x, y, z)
	return fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * 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 z around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 68.9

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in x around 0

   \[\leadsto {y}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto {y}^{\color{blue}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto e^{x} \]
10. Step-by-step derivation

11. Applied rewrites 51.3%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

    \[\leadsto 1 + x \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)} \]
13. Step-by-step derivation

14. Applied rewrites 28.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right) \]
15. Add Preprocessing

Alternative 9: 28.1% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (fma 0.5 x 1.0) x 1.0))

C

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

Julia

function code(x, y, z)
	return fma(fma(0.5, x, 1.0), x, 1.0)
end

Wolfram

code[x_, y_, z_] := N[(N[(0.5 * x + 1.0), $MachinePrecision] * 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 z around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 68.9

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in x around 0

   \[\leadsto {y}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto {y}^{\color{blue}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto e^{x} \]
10. Step-by-step derivation

11. Applied rewrites 51.3%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

    \[\leadsto 1 + x \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot x\right)} \]
13. Step-by-step derivation

14. Applied rewrites 26.5%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right) \]
15. Add Preprocessing

Alternative 10: 14.3% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(1.0 + x), $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 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 68.9

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in x around 0

   \[\leadsto {y}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto {y}^{\color{blue}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto e^{x} \]
10. Step-by-step derivation

11. Applied rewrites 51.3%

    \[\leadsto e^{x} \]

12. Taylor expanded in x around 0

    \[\leadsto 1 + x \]
13. Step-by-step derivation

14. Applied rewrites 13.2%

    \[\leadsto 1 + x \]
15. Add Preprocessing

Alternative 11: 14.1% accurate, 212.0× 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 z around 0

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

   1. +-commutative N/A

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

   2. exp-sum N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. exp-to-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 68.9

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

5. Applied rewrites 68.9%

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

6. Taylor expanded in x around 0

   \[\leadsto {y}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto {y}^{\color{blue}{y}} \]

9. Taylor expanded in y around 0

   \[\leadsto 1 \]
10. Step-by-step derivation

11. Applied rewrites 13.0%

    \[\leadsto 1 \]
12. Add Preprocessing

Developer Target 1: 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 2024296 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (exp (+ (- x z) (* (log y) y))))

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