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

Percentage Accurate: 100.0% → 100.0%

Time: 30.5s

Alternatives: 15

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: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(y \cdot x\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* (* y x) y))))
   (if (<= t_0 0.0)
     (exp (* y x))
     (if (<= t_0 5e+36) (fma (* y x) y 1.0) (exp y)))))

C

double code(double x, double y) {
	double t_0 = exp(((y * x) * y));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = exp((y * x));
	} else if (t_0 <= 5e+36) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = exp(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(Float64(y * x) * y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = exp(Float64(y * x));
	elseif (t_0 <= 5e+36)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = exp(y);
	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, 0.0], N[Exp[N[(y * x), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 5e+36], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[Exp[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 50.3%

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

   if 0.0 < (exp.f64 (*.f64 (*.f64 x y) y)) < 4.99999999999999977e36

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 97.4

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

   5. Applied rewrites 97.4%

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

   if 4.99999999999999977e36 < (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 38.5%

      \[\leadsto e^{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;e^{y \cdot x}\\ \mathbf{elif}\;e^{\left(y \cdot x\right) \cdot y} \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(y \cdot x\right) \cdot y}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (exp (* (* y x) y))))
   (if (<= t_0 0.0) (exp x) (if (<= t_0 5e+36) (fma (* y x) y 1.0) (exp y)))))

C

double code(double x, double y) {
	double t_0 = exp(((y * x) * y));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = exp(x);
	} else if (t_0 <= 5e+36) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = exp(y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = exp(Float64(Float64(y * x) * y))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = exp(x);
	elseif (t_0 <= 5e+36)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = exp(y);
	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, 0.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+36], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[Exp[y], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 0.0

   1. Initial program 100.0%

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

   4. Applied rewrites 61.8%

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

   if 0.0 < (exp.f64 (*.f64 (*.f64 x y) y)) < 4.99999999999999977e36

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 97.4

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

   5. Applied rewrites 97.4%

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

   if 4.99999999999999977e36 < (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 38.5%

      \[\leadsto e^{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(y \cdot x\right) \cdot y} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;e^{\left(y \cdot x\right) \cdot y} \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+15}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+76}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -2e+15)
     (exp x)
     (if (<= t_0 5e+21)
       (fma (* y x) y 1.0)
       (if (<= t_0 2e+76)
         (exp x)
         (fma
          (fma (* (fma (* 0.16666666666666666 y) x 0.5) (* y y)) x y)
          x
          1.0))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -2e+15) {
		tmp = exp(x);
	} else if (t_0 <= 5e+21) {
		tmp = fma((y * x), y, 1.0);
	} else if (t_0 <= 2e+76) {
		tmp = exp(x);
	} else {
		tmp = fma(fma((fma((0.16666666666666666 * y), x, 0.5) * (y * y)), x, y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -2e+15)
		tmp = exp(x);
	elseif (t_0 <= 5e+21)
		tmp = fma(Float64(y * x), y, 1.0);
	elseif (t_0 <= 2e+76)
		tmp = exp(x);
	else
		tmp = fma(fma(Float64(fma(Float64(0.16666666666666666 * y), x, 0.5) * Float64(y * y)), x, y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+15], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+76], N[Exp[x], $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e15 or 5e21 < (*.f64 (*.f64 x y) y) < 2.0000000000000001e76

   1. Initial program 100.0%

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

   4. Applied rewrites 64.1%

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

   if -2e15 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 96.7

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

   5. Applied rewrites 96.7%

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

   if 2.0000000000000001e76 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 40.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 46.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right), x, y\right), x, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -2 \cdot 10^{+15}:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+76}:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[(y * x + 1.0), $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 y around 0

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

   5. Applied rewrites 63.4%

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

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

   1. Initial program 99.9%

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

   4. Applied rewrites 38.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.4

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

   7. Applied rewrites 10.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.0%

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

Alternative 6: 65.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 5e+21)
       (fma (* y x) y 1.0)
       (fma
        (fma (* (fma (* 0.16666666666666666 y) x 0.5) (* y y)) x y)
        x
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 5e+21) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma((fma((0.16666666666666666 * y), x, 0.5) * (y * y)), x, y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 5e+21)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(Float64(fma(Float64(0.16666666666666666 * y), x, 0.5) * Float64(y * y)), x, y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 95.3

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

   5. Applied rewrites 95.3%

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

   if 5e21 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 40.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right), x, y\right), x, 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot y, x, 0.5\right) \cdot \left(y \cdot y\right), x, y\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(0.16666666666666666 \cdot x, y, 0.5\right) \cdot y\right) \cdot x, y\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 5e+21)
       (fma (* y x) y 1.0)
       (fma
        (fma y (* (* (fma (* 0.16666666666666666 x) y 0.5) y) x) y)
        x
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 5e+21) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(fma(y, ((fma((0.16666666666666666 * x), y, 0.5) * y) * x), y), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 5e+21)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(fma(y, Float64(Float64(fma(Float64(0.16666666666666666 * x), y, 0.5) * y) * x), y), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(y * N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * y + 0.5), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 95.3

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

   5. Applied rewrites 95.3%

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

   if 5e21 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 40.2%

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

   12. Applied rewrites 28.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(0.16666666666666666 \cdot x, y, 0.5\right) \cdot y\right) \cdot x, y\right), x, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, \left(\mathsf{fma}\left(0.16666666666666666 \cdot x, y, 0.5\right) \cdot y\right) \cdot x, y\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 5e+21)
       (fma (* y x) y 1.0)
       (* (* (* (fma 0.16666666666666666 (* y x) 0.5) x) x) (* y y))))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 5e+21) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = ((fma(0.16666666666666666, (y * x), 0.5) * x) * x) * (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 5e+21)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, Float64(y * x), 0.5) * x) * x) * Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * N[(y * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 95.3

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

   5. Applied rewrites 95.3%

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

   if 5e21 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 40.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 24.8%

       \[\leadsto \left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, y \cdot x, 0.5\right) \cdot x\right) \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 100.0) 1.0 (if (<= t_0 2e+305) (* (* 0.5 y) y) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 100.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+305) {
		tmp = (0.5 * y) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * y
    if (t_0 <= (-4d+20)) then
        tmp = (0.5d0 * x) * x
    else if (t_0 <= 100.0d0) then
        tmp = 1.0d0
    else if (t_0 <= 2d+305) then
        tmp = (0.5d0 * y) * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 100.0) {
		tmp = 1.0;
	} else if (t_0 <= 2e+305) {
		tmp = (0.5 * y) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if t_0 <= -4e+20:
		tmp = (0.5 * x) * x
	elif t_0 <= 100.0:
		tmp = 1.0
	elif t_0 <= 2e+305:
		tmp = (0.5 * y) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 100.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+305)
		tmp = Float64(Float64(0.5 * y) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -4e+20)
		tmp = (0.5 * x) * x;
	elseif (t_0 <= 100.0)
		tmp = 1.0;
	elseif (t_0 <= 2e+305)
		tmp = (0.5 * y) * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 100.0], 1.0, If[LessEqual[t$95$0, 2e+305], N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 100

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 95.4%

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

   if 100 < (*.f64 (*.f64 x y) y) < 1.9999999999999999e305

   1. Initial program 100.0%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 21.9

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

   7. Applied rewrites 21.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 21.8%

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

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{1}{2} \cdot y\right) \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 34.3%

       \[\leadsto \left(0.5 \cdot y\right) \cdot y \]

   if 1.9999999999999999e305 < (*.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 y 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 100:\\ \;\;\;\;1\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 5e+21) (fma (* y x) y 1.0) (* (* y y) x)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 5e+21) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 5e+21)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 5e+21], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 95.3

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

   5. Applied rewrites 95.3%

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

   if 5e21 < (*.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 y 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 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20) (* (* 0.5 x) x) (if (<= t_0 0.2) 1.0 (* (* y y) x)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 0.2) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * y
    if (t_0 <= (-4d+20)) then
        tmp = (0.5d0 * x) * x
    else if (t_0 <= 0.2d0) 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 t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 0.2) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if t_0 <= -4e+20:
		tmp = (0.5 * x) * x
	elif t_0 <= 0.2:
		tmp = 1.0
	else:
		tmp = (y * y) * x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 0.2)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -4e+20)
		tmp = (0.5 * x) * x;
	elseif (t_0 <= 0.2)
		tmp = 1.0;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 0.2], 1.0, N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 0.20000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 96.1%

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

   if 0.20000000000000001 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y 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 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 0.2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)))
   (if (<= t_0 -4e+20)
     (* (* 0.5 x) x)
     (if (<= t_0 100.0) 1.0 (* (* 0.5 y) y)))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 100.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * x) * y
    if (t_0 <= (-4d+20)) then
        tmp = (0.5d0 * x) * x
    else if (t_0 <= 100.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = (0.5 * x) * x;
	} else if (t_0 <= 100.0) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	tmp = 0
	if t_0 <= -4e+20:
		tmp = (0.5 * x) * x
	elif t_0 <= 100.0:
		tmp = 1.0
	else:
		tmp = (0.5 * y) * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = Float64(Float64(0.5 * x) * x);
	elseif (t_0 <= 100.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	tmp = 0.0;
	if (t_0 <= -4e+20)
		tmp = (0.5 * x) * x;
	elseif (t_0 <= 100.0)
		tmp = 1.0;
	else
		tmp = (0.5 * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, 100.0], 1.0, N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20

   1. Initial program 100.0%

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

   4. Applied rewrites 60.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 2.4

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

   7. Applied rewrites 2.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 18.3%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 100

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 95.4%

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

   if 100 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 38.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 26.6

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

   7. Applied rewrites 26.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 26.5%

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

   11. Taylor expanded in y around 0

       \[\leadsto \left(\frac{1}{2} \cdot y\right) \cdot y \]
   12. Step-by-step derivation

   13. Applied rewrites 53.0%

       \[\leadsto \left(0.5 \cdot y\right) \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 100:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot x\right) \cdot y\\ t_1 := \left(0.5 \cdot x\right) \cdot x\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y x) y)) (t_1 (* (* 0.5 x) x)))
   (if (<= t_0 -4e+20) t_1 (if (<= t_0 5e+21) 1.0 t_1))))

C

double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (0.5 * x) * x;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = t_1;
	} else if (t_0 <= 5e+21) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y * x) * y
    t_1 = (0.5d0 * x) * x
    if (t_0 <= (-4d+20)) then
        tmp = t_1
    else if (t_0 <= 5d+21) then
        tmp = 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * x) * y;
	double t_1 = (0.5 * x) * x;
	double tmp;
	if (t_0 <= -4e+20) {
		tmp = t_1;
	} else if (t_0 <= 5e+21) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * x) * y
	t_1 = (0.5 * x) * x
	tmp = 0
	if t_0 <= -4e+20:
		tmp = t_1
	elif t_0 <= 5e+21:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * x) * y)
	t_1 = Float64(Float64(0.5 * x) * x)
	tmp = 0.0
	if (t_0 <= -4e+20)
		tmp = t_1;
	elseif (t_0 <= 5e+21)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * x) * y;
	t_1 = (0.5 * x) * x;
	tmp = 0.0;
	if (t_0 <= -4e+20)
		tmp = t_1;
	elseif (t_0 <= 5e+21)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+20], t$95$1, If[LessEqual[t$95$0, 5e+21], 1.0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e20 or 5e21 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 19.0

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

   7. Applied rewrites 19.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{{x}^{2}} \]
   9. Step-by-step derivation

   10. Applied rewrites 27.5%

       \[\leadsto \left(x \cdot 0.5\right) \cdot \color{blue}{x} \]

   if -4e20 < (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \mathbf{elif}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* (* y x) y) 5e+21) 1.0 (* y x)))

C

double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 5e+21) {
		tmp = 1.0;
	} else {
		tmp = 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 (((y * x) * y) <= 5d+21) then
        tmp = 1.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * x) * y) <= 5e+21) {
		tmp = 1.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * x) * y) <= 5e+21:
		tmp = 1.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y * x) * y) <= 5e+21)
		tmp = 1.0;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * x) * y) <= 5e+21)
		tmp = 1.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision], 5e+21], 1.0, N[(y * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5e21

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 62.8%

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

   if 5e21 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 39.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 10.4%

       \[\leadsto x \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot x\right) \cdot y \leq 5 \cdot 10^{+21}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.2% 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 y around 0

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

5. Applied rewrites 49.3%

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

Reproduce

herbie shell --seed 2024268 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))