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

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.0×

• 25.2% 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: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(x \cdot y\right) \cdot y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(y \cdot y\right) \cdot x\right) \cdot x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* (* x y) y)) 2.0) 1.0 (* (sqrt (* (* (* y y) x) x)) y)))

C

double code(double x, double y) {
	double tmp;
	if (exp(((x * y) * y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt((((y * y) * x) * x)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (exp(((x * y) * y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((((y * y) * x) * x)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.exp(((x * y) * y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((((y * y) * x) * x)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp(((x * y) * y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = math.sqrt((((y * y) * x) * x)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(Float64(x * y) * y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(y * y) * x) * x)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (exp(((x * y) * y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt((((y * y) * x) * x)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[(N[Sqrt[N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (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

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

   5. Applied rewrites 73.8%

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

   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

   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 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.0%

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

   10. Applied rewrites 42.1%

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

Alternative 3: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(x \cdot y\right) \cdot y} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* (* x y) y)) 2.0) 1.0 (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if (exp(((x * y) * y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (exp(((x * y) * y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.exp(((x * y) * y)) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp(((x * y) * y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = (y * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(Float64(x * y) * y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (exp(((x * y) * y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (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

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

   5. Applied rewrites 73.8%

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

   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

   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 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.0%

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

Alternative 4: 72.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \left(\left(\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, 0.5\right) \cdot y\right) \cdot y\right), x\right) \cdot y, y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  (* (fma x (* x (* (* (fma 0.16666666666666666 (* (* y y) x) 0.5) y) y)) x) y)
  y
  1.0))

C

double code(double x, double y) {
	return fma((fma(x, (x * ((fma(0.16666666666666666, ((y * y) * x), 0.5) * y) * y)), x) * y), y, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(fma(x, Float64(x * Float64(Float64(fma(0.16666666666666666, Float64(Float64(y * y) * x), 0.5) * y) * y)), x) * y), y, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(x * N[(N[(N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * y), $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

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, 0.5\right), y \cdot y, x\right) \cdot y, y, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 77.3%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \left(\left(\mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, 0.5\right) \cdot y\right) \cdot y\right), x\right) \cdot y, y, 1\right) \]
11. Add Preprocessing

Alternative 5: 72.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, \left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(y \cdot y\right), x\right) \cdot y, y, 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  (* (fma x (* (* (* (* (* y y) x) x) 0.16666666666666666) (* y y)) x) y)
  y
  1.0))

C

double code(double x, double y) {
	return fma((fma(x, (((((y * y) * x) * x) * 0.16666666666666666) * (y * y)), x) * y), y, 1.0);
}

Julia

function code(x, y)
	return fma(Float64(fma(x, Float64(Float64(Float64(Float64(Float64(y * y) * x) * x) * 0.16666666666666666) * Float64(y * y)), x) * y), y, 1.0)
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] * y), $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

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, 0.5\right), y \cdot y, x\right) \cdot y, y, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 76.6%

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

11. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \left(\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{2}\right)\right) \cdot \left(y \cdot y\right), x\right) \cdot y, y, 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 76.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \left(\left(\left(\left(y \cdot y\right) \cdot x\right) \cdot x\right) \cdot 0.16666666666666666\right) \cdot \left(y \cdot y\right), x\right) \cdot y, y, 1\right) \]
14. Add Preprocessing

Alternative 6: 70.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 54.5%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 67.1%

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

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\left(x \cdot \left(1 + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)\right) \cdot y, y, 1\right) \]
10. Step-by-step derivation

11. Applied rewrites 76.1%

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

Alternative 7: 62.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 2 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* x y) y) 2e-14) 1.0 (* (* y x) y)))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 2e-14) {
		tmp = 1.0;
	} else {
		tmp = (y * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * y) * y) <= 2d-14) then
        tmp = 1.0d0
    else
        tmp = (y * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 2e-14) {
		tmp = 1.0;
	} else {
		tmp = (y * x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * y) * y) <= 2e-14:
		tmp = 1.0
	else:
		tmp = (y * x) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 2e-14)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * y) * y) <= 2e-14)
		tmp = 1.0;
	else
		tmp = (y * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 2e-14], 1.0, N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 2e-14

   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 73.8%

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

   if 2e-14 < (*.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 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 69.0%

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

   10. Applied rewrites 57.0%

       \[\leadsto \left(y \cdot x\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.1% 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 72.4

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

5. Applied rewrites 72.4%

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

Alternative 9: 50.8% 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 54.5%

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

Reproduce

herbie shell --seed 2024329 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))