Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2

Percentage Accurate: 100.0% → 100.0%

Time: 25.4s

Alternatives: 11

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* (* x y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* (* x y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* (* y x) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return exp(Float64(Float64(y * x) * y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 71.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(y \cdot x\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-153}:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* (* y x) y))))
   (if (<= t_0 4e-153)
     (exp (* y x))
     (if (<= t_0 2.0)
       (fma (* y x) y 1.0)
       (* (* (* (* (* (* y y) 0.16666666666666666) y) x) x) x)))))

C

double code(double x, double y) {
	double t_0 = exp(((y * x) * y));
	double tmp;
	if (t_0 <= 4e-153) {
		tmp = exp((y * x));
	} else if (t_0 <= 2.0) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (((((y * y) * 0.16666666666666666) * y) * x) * x) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(Float64(y * x) * y))
	tmp = 0.0
	if (t_0 <= 4e-153)
		tmp = exp(Float64(y * x));
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * y) * x) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Exp[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 4e-153], N[Exp[N[(y * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 4.00000000000000016e-153

   1. Initial program 100.0%

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

   4. Applied rewrites 42.1%

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

   if 4.00000000000000016e-153 < (exp.f64 (*.f64 (*.f64 x y) y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot {y}^{2} + 1} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot y\right)} + 1 \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot y} + 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, y, 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot x}, y, 1\right) \]

      6. lower-*.f64 99.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot x}, y, 1\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, y, 1\right)} \]

   if 2 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right) + 1} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot y} + 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x + y \cdot \left(\frac{1}{6} \cdot \left({x}^{3} \cdot y\right) + \frac{1}{2} \cdot {x}^{2}\right), y, 1\right)} \]

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right) \cdot x, y, \left(0.5 \cdot x\right) \cdot x\right), y, x\right), y, 1\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left({x}^{3} \cdot {y}^{3}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.4%

       \[\leadsto \left(\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 44.4%

       \[\leadsto \left(\left(\left(\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot y\right) \cdot x\right) \cdot x\right) \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 4 \cdot 10^{-153}:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;e^{\left(y \cdot x\right) \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\right) \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))