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

Percentage Accurate: 100.0% → 100.0%

Time: 8.3s

Alternatives: 7

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

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

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: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+137}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) y))) (if (<= t_0 5e+137) (exp (- x z)) (exp t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * y;
	double tmp;
	if (t_0 <= 5e+137) {
		tmp = exp((x - z));
	} else {
		tmp = exp(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 = log(y) * y
    if (t_0 <= 5d+137) then
        tmp = exp((x - z))
    else
        tmp = exp(t_0)
    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 <= 5e+137) {
		tmp = Math.exp((x - z));
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) * y
	tmp = 0
	if t_0 <= 5e+137:
		tmp = math.exp((x - z))
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * y)
	tmp = 0.0
	if (t_0 <= 5e+137)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) * y;
	tmp = 0.0;
	if (t_0 <= 5e+137)
		tmp = exp((x - z));
	else
		tmp = exp(t_0);
	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, 5e+137], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 5.0000000000000002e137

   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 92.7

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

   5. Applied rewrites 92.7%

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

   if 5.0000000000000002e137 < (*.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 y around inf

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{y}\right) \cdot y}\right)} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto e^{\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right) \cdot y}} \]

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 91.1

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

   5. Applied rewrites 91.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 5 \cdot 10^{+137}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 6.6 \cdot 10^{+214}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(z, z - x, x \cdot x\right) \cdot \mathsf{fma}\left(x, \frac{x}{\left(z \cdot z\right) \cdot z}, \frac{-1}{z}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) 6.6e+214)
   (exp (- x z))
   (exp (* (fma z (- z x) (* x x)) (fma x (/ x (* (* z z) z)) (/ -1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= 6.6e+214) {
		tmp = exp((x - z));
	} else {
		tmp = exp((fma(z, (z - x), (x * x)) * fma(x, (x / ((z * z) * z)), (-1.0 / z))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= 6.6e+214)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(Float64(fma(z, Float64(z - x), Float64(x * x)) * fma(x, Float64(x / Float64(Float64(z * z) * z)), Float64(-1.0 / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], 6.6e+214], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(z * N[(z - x), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 6.60000000000000023e214

   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 87.6

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

   5. Applied rewrites 87.6%

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

   if 6.60000000000000023e214 < (*.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 y around 0

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

      1. lower--.f64 44.1

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

   5. Applied rewrites 44.1%

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

   7. Applied rewrites 15.5%

      \[\leadsto e^{\frac{\left(z + x\right) \cdot \left(x - z\right)}{\mathsf{fma}\left(z \cdot z, z, \left(x \cdot x\right) \cdot x\right)} \cdot \color{blue}{\mathsf{fma}\left(z, z - x, x \cdot x\right)}} \]

   8. Taylor expanded in x around 0

      \[\leadsto e^{\left(\frac{{x}^{2}}{{z}^{3}} - \frac{1}{z}\right) \cdot \mathsf{fma}\left(\color{blue}{z}, z - x, x \cdot x\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.9%

       \[\leadsto e^{\mathsf{fma}\left(x, \frac{x}{\left(z \cdot z\right) \cdot z}, \frac{-1}{z}\right) \cdot \mathsf{fma}\left(\color{blue}{z}, z - x, x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 6.6 \cdot 10^{+214}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(z, z - x, x \cdot x\right) \cdot \mathsf{fma}\left(x, \frac{x}{\left(z \cdot z\right) \cdot z}, \frac{-1}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 2.4 \cdot 10^{+264}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z}, -\left(z - x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* (log y) y) 2.4e+264)
   (exp (- x z))
   (exp (fma (* x x) (/ 1.0 z) (- (- z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((log(y) * y) <= 2.4e+264) {
		tmp = exp((x - z));
	} else {
		tmp = exp(fma((x * x), (1.0 / z), -(z - x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(log(y) * y) <= 2.4e+264)
		tmp = exp(Float64(x - z));
	else
		tmp = exp(fma(Float64(x * x), Float64(1.0 / z), Float64(-Float64(z - x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision], 2.4e+264], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(x * x), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + (-N[(z - x), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (log.f64 y)) < 2.39999999999999993e264

   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 84.5

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

   5. Applied rewrites 84.5%

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

   if 2.39999999999999993e264 < (*.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 y around 0

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

      1. lower--.f64 34.7

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

   5. Applied rewrites 34.7%

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

   7. Applied rewrites 34.7%

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{z + x}}, -z \cdot \frac{z}{z + x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, \mathsf{neg}\left(\left(z + -1 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 34.7%

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, -\left(z - x\right)\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{\color{blue}{z}}, \mathsf{neg}\left(\left(z - x\right)\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 56.4%

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{\color{blue}{z}}, -\left(z - x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 2.4 \cdot 10^{+264}:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z}, -\left(z - x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x - z}\\ \mathbf{if}\;z \leq 5.2 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-156}:\\ \;\;\;\;e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z}, -\left(z - x\right)\right)}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;e^{\mathsf{fma}\left(\frac{x}{z + x}, x, \frac{z - \frac{z \cdot z}{x}}{x} \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (exp (- x z))))
   (if (<= z 5.2e-307)
     t_0
     (if (<= z 2.45e-156)
       (exp (fma (* x x) (/ 1.0 z) (- (- z x))))
       (if (<= z 5e+80)
         (exp (fma (/ x (+ z x)) x (* (/ (- z (/ (* z z) x)) x) (- z))))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = exp((x - z));
	double tmp;
	if (z <= 5.2e-307) {
		tmp = t_0;
	} else if (z <= 2.45e-156) {
		tmp = exp(fma((x * x), (1.0 / z), -(z - x)));
	} else if (z <= 5e+80) {
		tmp = exp(fma((x / (z + x)), x, (((z - ((z * z) / x)) / x) * -z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = exp(Float64(x - z))
	tmp = 0.0
	if (z <= 5.2e-307)
		tmp = t_0;
	elseif (z <= 2.45e-156)
		tmp = exp(fma(Float64(x * x), Float64(1.0 / z), Float64(-Float64(z - x))));
	elseif (z <= 5e+80)
		tmp = exp(fma(Float64(x / Float64(z + x)), x, Float64(Float64(Float64(z - Float64(Float64(z * z) / x)) / x) * Float64(-z))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z, 5.2e-307], t$95$0, If[LessEqual[z, 2.45e-156], N[Exp[N[(N[(x * x), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + (-N[(z - x), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[z, 5e+80], N[Exp[N[(N[(x / N[(z + x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(z - N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * (-z)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.19999999999999992e-307 or 4.99999999999999961e80 < 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 90.9

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

   5. Applied rewrites 90.9%

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

   if 5.19999999999999992e-307 < z < 2.44999999999999976e-156

   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 51.6

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

   5. Applied rewrites 51.6%

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

   7. Applied rewrites 51.6%

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{z + x}}, -z \cdot \frac{z}{z + x}\right)} \]

   8. Taylor expanded in x around 0

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, \mathsf{neg}\left(\left(z + -1 \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.1%

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, -\left(z - x\right)\right)} \]

   11. Taylor expanded in z around inf

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{\color{blue}{z}}, \mathsf{neg}\left(\left(z - x\right)\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.7%

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{\color{blue}{z}}, -\left(z - x\right)\right)} \]

   if 2.44999999999999976e-156 < z < 4.99999999999999961e80

   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 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.1%

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{z + x}}, -z \cdot \frac{z}{z + x}\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, \mathsf{neg}\left(z \cdot \left(z \cdot \left(-1 \cdot \frac{z}{{x}^{2}} + \frac{1}{x}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.1%

       \[\leadsto e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z + x}, -z \cdot \frac{z - \frac{z \cdot z}{x}}{x}\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.1%

       \[\leadsto e^{\mathsf{fma}\left(\frac{x}{z + x}, \color{blue}{x}, \left(-z\right) \cdot \frac{z - \frac{z \cdot z}{x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-307}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-156}:\\ \;\;\;\;e^{\mathsf{fma}\left(x \cdot x, \frac{1}{z}, -\left(z - x\right)\right)}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;e^{\mathsf{fma}\left(\frac{x}{z + x}, x, \frac{z - \frac{z \cdot z}{x}}{x} \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{x - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.2

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

5. Applied rewrites 79.2%

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

Alternative 7: 52.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{-z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(-z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.7

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

5. Applied rewrites 49.7%

   \[\leadsto e^{\color{blue}{-z}} \]
6. 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 2024235 
(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)))