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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 6

Speedup: 1.0×

• 24.9% 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

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 72.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 2 \cdot 10^{-31}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \left(y \cdot \left(x \cdot x\right)\right), 0.5, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* x y) y) 2e-31)
   1.0
   (fma (fma (* y (* y (* x x))) 0.5 x) (* y y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 2e-31) {
		tmp = 1.0;
	} else {
		tmp = fma(fma((y * (y * (x * x))), 0.5, x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 2e-31)
		tmp = 1.0;
	else
		tmp = fma(fma(Float64(y * Float64(y * Float64(x * x))), 0.5, x), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 2e-31], 1.0, N[(N[(N[(y * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e-31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.2%

      \[\leadsto \color{blue}{1} \]

   if 2e-31 < (*.f64 (*.f64 x y) y)

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 73.3

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

   5. Applied rewrites 73.3%

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

   7. Applied rewrites 65.3%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 84.1

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

   10. Applied rewrites 84.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x, 0.5, x\right), y \cdot y, 1\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 88.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y \cdot \left(y \cdot \left(x \cdot x\right)\right), 0.5, x\right), y \cdot y, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x, 0.5, x\right), y \cdot y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma (fma (* (* (* y y) x) x) 0.5 x) (* y y) 1.0))

C

double code(double x, double y) {
	return fma(fma((((y * y) * x) * x), 0.5, x), (y * y), 1.0);
}

Julia

function code(x, y)
	return fma(fma(Float64(Float64(Float64(y * y) * x) * x), 0.5, x), Float64(y * y), 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 67.3

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

5. Applied rewrites 67.3%

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

7. Applied rewrites 65.1%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 70.3

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

10. Applied rewrites 70.3%

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

Alternative 4: 67.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot y, x, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* y y) x 1.0))

C

double code(double x, double y) {
	return fma((y * y), x, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(y * y), x, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 67.3

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

5. Applied rewrites 67.3%

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

Alternative 5: 63.8% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot x, y, 1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* y x) y 1.0))

C

double code(double x, double y) {
	return fma((y * x), y, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(y * x), y, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision]

Derivation

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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 67.3

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

5. Applied rewrites 67.3%

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

7. Applied rewrites 65.1%

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

Alternative 6: 51.5% accurate, 111.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))