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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 4

Speedup: 1.0×

• 57.4% 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: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y \cdot y\\ t_1 := e^{t\_0}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+54}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+204}:\\ \;\;\;\;e^{\frac{\mathsf{fma}\left(-z, z, x \cdot x\right)}{z + x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (log y) y)) (t_1 (exp t_0)))
   (if (<= t_0 5e+54)
     (exp (- x z))
     (if (<= t_0 5e+141)
       t_1
       (if (<= t_0 2e+204) (exp (/ (fma (- z) z (* x x)) (+ z x))) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) * y;
	double t_1 = exp(t_0);
	double tmp;
	if (t_0 <= 5e+54) {
		tmp = exp((x - z));
	} else if (t_0 <= 5e+141) {
		tmp = t_1;
	} else if (t_0 <= 2e+204) {
		tmp = exp((fma(-z, z, (x * x)) / (z + x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(y) * y)
	t_1 = exp(t_0)
	tmp = 0.0
	if (t_0 <= 5e+54)
		tmp = exp(Float64(x - z));
	elseif (t_0 <= 5e+141)
		tmp = t_1;
	elseif (t_0 <= 2e+204)
		tmp = exp(Float64(fma(Float64(-z), z, Float64(x * x)) / Float64(z + x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, If[LessEqual[t$95$0, 5e+54], N[Exp[N[(x - z), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+141], t$95$1, If[LessEqual[t$95$0, 2e+204], N[Exp[N[(N[((-z) * z + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(z + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (log.f64 y)) < 5.00000000000000005e54

   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 94.3

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

   5. Applied rewrites 94.3%

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

   if 5.00000000000000005e54 < (*.f64 y (log.f64 y)) < 5.00000000000000025e141 or 1.99999999999999998e204 < (*.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 88.4

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

   5. Applied rewrites 88.4%

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

   if 5.00000000000000025e141 < (*.f64 y (log.f64 y)) < 1.99999999999999998e204

   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 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 81.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot y \leq 5 \cdot 10^{+54}:\\ \;\;\;\;e^{x - z}\\ \mathbf{elif}\;\log y \cdot y \leq 5 \cdot 10^{+141}:\\ \;\;\;\;e^{\log y \cdot y}\\ \mathbf{elif}\;\log y \cdot y \leq 2 \cdot 10^{+204}:\\ \;\;\;\;e^{\frac{\mathsf{fma}\left(-z, z, x \cdot x\right)}{z + x}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% 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 79.5

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

5. Applied rewrites 79.5%

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

Alternative 4: 52.8% 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 52.2

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

5. Applied rewrites 52.2%

   \[\leadsto e^{\color{blue}{-z}} \]
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 2024249 
(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)))