Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 10.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(-z\right) - y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (log y) x (- (- z) y)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot \log y - z\right) - y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y - z\right) - y} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y - z\right)} - y \]

   3. sub-neg N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(z\right)\right)\right)} - y \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right) \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(z\right)\right) - y\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) - y}\right) \]

   9. lower-neg.f64 99.9

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-z\right)} - y\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-z\right) - y\right)} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-z\right) - y\\ \mathbf{if}\;z \leq -6.8 \cdot 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (- z) y)))
   (if (<= z -6.8e+130) t_0 (if (<= z 2.7e+118) (fma (log y) x (- y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -z - y;
	double tmp;
	if (z <= -6.8e+130) {
		tmp = t_0;
	} else if (z <= 2.7e+118) {
		tmp = fma(log(y), x, -y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) - y)
	tmp = 0.0
	if (z <= -6.8e+130)
		tmp = t_0;
	elseif (z <= 2.7e+118)
		tmp = fma(log(y), x, Float64(-y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) - y), $MachinePrecision]}, If[LessEqual[z, -6.8e+130], t$95$0, If[LessEqual[z, 2.7e+118], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000001e130 or 2.7e118 < z

   1. Initial program 99.9%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - y \]

      2. lower-neg.f64 85.7

         \[\leadsto \color{blue}{\left(-z\right)} - y \]

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\left(-z\right)} - y \]

   if -6.8000000000000001e130 < z < 2.7e118

   1. Initial program 99.8%

      \[\left(x \cdot \log y - z\right) - y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - z\right) - y} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - z\right)} - y \]

      3. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(z\right)\right)\right)} - y \]

      4. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(z\right)\right) - y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(z\right)\right) - y\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right) - y}\right) \]

      9. lower-neg.f64 99.8

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-z\right)} - y\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-z\right) - y\right)} \]

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot y}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \]

      2. lower-neg.f64 82.9

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]

   7. Applied rewrites 82.9%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson:$clogProbability from math-functions-0.1.5.2"
  :precision binary64
  (- (- (* x (log y)) z) y))