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

Percentage Accurate: 100.0% → 100.0%

Time: 8.9s

Alternatives: 5

Speedup: 1.0×

• 57.8% 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: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;t\_0 \leq 6.6 \cdot 10^{+192}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;t\_0 \leq 6.5 \cdot 10^{+269}:\\ \;\;\;\;e^{\mathsf{fma}\left(z, x + z, x \cdot x\right) \cdot \frac{-1}{z}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(z \cdot \left(z \cdot \left(1 + \mathsf{fma}\left(x, \frac{x}{z \cdot z}, \frac{x}{z}\right)\right)\right)\right) \cdot \frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= t_0 6.6e+192)
     (exp (- x z))
     (if (<= t_0 6.5e+269)
       (exp (* (fma z (+ x z) (* x x)) (/ -1.0 z)))
       (exp
        (* (* z (* z (+ 1.0 (fma x (/ x (* z z)) (/ x z))))) (/ 1.0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (t_0 <= 6.6e+192) {
		tmp = exp((x - z));
	} else if (t_0 <= 6.5e+269) {
		tmp = exp((fma(z, (x + z), (x * x)) * (-1.0 / z)));
	} else {
		tmp = exp(((z * (z * (1.0 + fma(x, (x / (z * z)), (x / z))))) * (1.0 / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (t_0 <= 6.6e+192)
		tmp = exp(Float64(x - z));
	elseif (t_0 <= 6.5e+269)
		tmp = exp(Float64(fma(z, Float64(x + z), Float64(x * x)) * Float64(-1.0 / z)));
	else
		tmp = exp(Float64(Float64(z * Float64(z * Float64(1.0 + fma(x, Float64(x / Float64(z * z)), Float64(x / z))))) * Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 6.6e+192], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 6.5e+269], N[Exp[N[(N[(z * N[(x + z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(z * N[(z * N[(1.0 + N[(x * N[(x / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < 6.60000000000000019e192

   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 87.0

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

   5. Applied rewrites 87.0%

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

   if 6.60000000000000019e192 < (*.f64 y (log.f64 y)) < 6.5000000000000003e269

   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 53.3

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

   5. Applied rewrites 53.3%

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

   7. Applied rewrites 18.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 47.3%

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

   if 6.5000000000000003e269 < (*.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 43.8

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

   5. Applied rewrites 43.8%

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

   7. Applied rewrites 16.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 45.5%

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

   11. Taylor expanded in z around inf

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

   13. Applied rewrites 46.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \log y \leq 6.6 \cdot 10^{+192}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;y \cdot \log y \leq 6.5 \cdot 10^{+269}:\\ \;\;\;\;e^{\mathsf{fma}\left(z, x + z, x \cdot x\right) \cdot \frac{-1}{z}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(z \cdot \left(z \cdot \left(1 + \mathsf{fma}\left(x, \frac{x}{z \cdot z}, \frac{x}{z}\right)\right)\right)\right) \cdot \frac{1}{x}}\\ \end{array} \]
5. 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 2024228 
(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)))