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

Percentage Accurate: 100.0% → 100.0%

Time: 36.8s

Alternatives: 20

Speedup: 1.0×

• 24.2% 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

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

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(y * Float64(x * y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 82.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 0:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 0.0)
   (exp (* x y))
   (fma
    (* (* y y) (* (* y y) (* x (fma (* x (* y y)) 0.16666666666666666 0.5))))
    x
    (fma x (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 0.0) {
		tmp = exp((x * y));
	} else {
		tmp = fma(((y * y) * ((y * y) * (x * fma((x * (y * y)), 0.16666666666666666, 0.5)))), x, fma(x, (y * y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 0.0)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x * fma(Float64(x * Float64(y * y)), 0.16666666666666666, 0.5)))), x, fma(x, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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 egg-rr 44.9%

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

   if 0.0 < (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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Simplified 92.0%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right) \cdot \left(y \cdot y\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 0:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 0.0)
   (exp x)
   (fma
    (* (* y y) (* (* y y) (* x (fma (* x (* y y)) 0.16666666666666666 0.5))))
    x
    (fma x (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 0.0) {
		tmp = exp(x);
	} else {
		tmp = fma(((y * y) * ((y * y) * (x * fma((x * (y * y)), 0.16666666666666666, 0.5)))), x, fma(x, (y * y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 0.0)
		tmp = exp(x);
	else
		tmp = fma(Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x * fma(Float64(x * Float64(y * y)), 0.16666666666666666, 0.5)))), x, fma(x, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Exp[x], $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 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 egg-rr 71.1%

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

   if 0.0 < (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

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Simplified 92.0%

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   7. Applied egg-rr 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right) \cdot \left(y \cdot y\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(y \cdot y\right), 0.16666666666666666, 0.5\right)\right)\right), x, \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot y\right) \cdot \mathsf{fma}\left(x \cdot y, y \cdot 0.5, 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, x \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2.0)
   (fma (* (* x y) (fma (* x y) (* y 0.5) 1.0)) y 1.0)
   (fma x (fma x (* (* y y) (fma y (* x 0.16666666666666666) 0.5)) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		tmp = fma(((x * y) * fma((x * y), (y * 0.5), 1.0)), y, 1.0);
	} else {
		tmp = fma(x, fma(x, ((y * y) * fma(y, (x * 0.16666666666666666), 0.5)), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		tmp = fma(Float64(Float64(x * y) * fma(Float64(x * y), Float64(y * 0.5), 1.0)), y, 1.0);
	else
		tmp = fma(x, fma(x, Float64(Float64(y * y) * fma(y, Float64(x * 0.16666666666666666), 0.5)), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(N[(x * y), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 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 x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 67.6%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 67.6

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

   7. Applied egg-rr 67.6%

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

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 40.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 38.9

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

   7. Simplified 38.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \mathsf{fma}\left(y, x \cdot 0.16666666666666666, 0.5\right), y\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.5%

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

Alternative 5: 60.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2.0)
   (fma (* x y) y 1.0)
   (* y (* 0.16666666666666666 (* x (* x (* x (* y y))))))))

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(y * Float64(0.16666666666666666 * Float64(x * Float64(x * Float64(x * Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(y * N[(0.16666666666666666 * N[(x * N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 67.5

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

   7. Applied egg-rr 67.5%

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

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 40.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 29.0

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

   10. Simplified 29.0%

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

   11. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

       17. associate-*l* N/A

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

       18. *-lowering-*.f64 N/A

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

       19. unpow2 N/A

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

       20. associate-*l* N/A

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

       21. *-lowering-*.f64 N/A

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

       22. *-lowering-*.f64 N/A

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 35.2

           \[\leadsto y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

   13. Simplified 35.2%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

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

Alternative 6: 71.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2.0)
   (fma (* x y) y 1.0)
   (* x (* 0.5 (* x (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 67.5

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

   7. Applied egg-rr 67.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 73.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({y}^{4} \cdot {x}^{2}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot {x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{4} \cdot x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 78.9

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

   8. Simplified 78.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

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

Alternative 7: 69.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2.0)
   (fma (* x y) y 1.0)
   (fma x (fma x (* (* y y) 0.5) y) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = fma(x, fma(x, ((y * y) * 0.5), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2.0)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = fma(x, fma(x, Float64(Float64(y * y) * 0.5), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 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 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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.4

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

   5. Simplified 67.4%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 67.5

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

   7. Applied egg-rr 67.5%

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

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 40.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 38.9

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

   7. Simplified 38.9%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 76.5

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

   10. Simplified 76.5%

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

4. Final simplification 70.3%

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

Alternative 8: 54.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   4. Applied egg-rr 67.4%

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

   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

   4. Applied egg-rr 40.7%

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

   5. Taylor expanded in x 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. accelerator-lowering-fma.f64 15.4

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

   7. Simplified 15.4%

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

4. Final simplification 51.0%

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

Alternative 9: 70.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot y\right) \cdot \mathsf{fma}\left(x \cdot y, y \cdot 0.5, 1\right), y, 1\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.5e-77)
   (fma (* (* x y) (fma (* x y) (* y 0.5) 1.0)) y 1.0)
   (if (<= y 1.4e+87)
     (fma
      (* y y)
      (* (* y y) (* (* y y) (* x (* 0.16666666666666666 (* x x)))))
      1.0)
     (* x (* 0.5 (* x (* (* y y) (* y y))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-77) {
		tmp = fma(((x * y) * fma((x * y), (y * 0.5), 1.0)), y, 1.0);
	} else if (y <= 1.4e+87) {
		tmp = fma((y * y), ((y * y) * ((y * y) * (x * (0.16666666666666666 * (x * x))))), 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-77)
		tmp = fma(Float64(Float64(x * y) * fma(Float64(x * y), Float64(y * 0.5), 1.0)), y, 1.0);
	elseif (y <= 1.4e+87)
		tmp = fma(Float64(y * y), Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(x * Float64(0.16666666666666666 * Float64(x * x))))), 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-77], N[(N[(N[(x * y), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[y, 1.4e+87], N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x * N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.50000000000000013e-77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 78.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 78.7

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

   7. Applied egg-rr 78.7%

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

   if 3.50000000000000013e-77 < y < 1.40000000000000008e87

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

   5. Simplified 53.1%

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

   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. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 60.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot {y}^{4}\right)}, 1\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{\left(\frac{1}{6} \cdot {x}^{3}\right) \cdot {y}^{4}}, 1\right) \]

       2. metadata-eval N/A

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

       3. pow-sqr N/A

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

       4. associate-*l* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. *-commutative N/A

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

       16. cube-mult N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

       20. *-lowering-*.f64 N/A

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

       21. *-commutative N/A

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

       22. *-lowering-*.f64 N/A

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 60.7

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

   11. Simplified 60.7%

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

   if 1.40000000000000008e87 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 53.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({y}^{4} \cdot {x}^{2}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot {x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{4} \cdot x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 59.0

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

   8. Simplified 59.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot y\right) \cdot \mathsf{fma}\left(x \cdot y, y \cdot 0.5, 1\right), y, 1\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+87}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $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

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 71.9%

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 72.6%

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

8. Final simplification 72.6%

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

Alternative 11: 60.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (if (<= (* y (* x y)) 1e+27)
     (fma t_0 (fma x (* (* y y) 0.5) 1.0) 1.0)
     (* y (* 0.16666666666666666 (* x (* x t_0)))))))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	double tmp;
	if ((y * (x * y)) <= 1e+27) {
		tmp = fma(t_0, fma(x, ((y * y) * 0.5), 1.0), 1.0);
	} else {
		tmp = y * (0.16666666666666666 * (x * (x * t_0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e+27)
		tmp = fma(t_0, fma(x, Float64(Float64(y * y) * 0.5), 1.0), 1.0);
	else
		tmp = Float64(y * Float64(0.16666666666666666 * Float64(x * Float64(x * t_0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e+27], N[(t$95$0 * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(y * N[(0.16666666666666666 * N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 64.8%

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

   if 1e27 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 40.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 42.8

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

   7. Simplified 42.8%

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

   8. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 31.8

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

   10. Simplified 31.8%

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

   11. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. unpow2 N/A

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

       8. associate-*r* N/A

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

       9. associate-*r* N/A

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

       10. *-lowering-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. associate-*r* N/A

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

       13. unpow2 N/A

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

       14. *-lowering-*.f64 N/A

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

       15. cube-mult N/A

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

       16. unpow2 N/A

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

       17. associate-*l* N/A

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

       18. *-lowering-*.f64 N/A

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

       19. unpow2 N/A

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

       20. associate-*l* N/A

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

       21. *-lowering-*.f64 N/A

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

       22. *-lowering-*.f64 N/A

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

       23. unpow2 N/A

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

       24. *-lowering-*.f64 38.8

           \[\leadsto y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

   13. Simplified 38.8%

       \[\leadsto \color{blue}{y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(y \cdot y\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot y\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, x \cdot \mathsf{fma}\left(t\_0, 0.16666666666666666, 0.5\right), 1\right), t\_0, 1\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (* y y))))
   (fma (fma (* y y) (* x (fma t_0 0.16666666666666666 0.5)) 1.0) t_0 1.0)))

C

double code(double x, double y) {
	double t_0 = x * (y * y);
	return fma(fma((y * y), (x * fma(t_0, 0.16666666666666666, 0.5)), 1.0), t_0, 1.0);
}

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y * y))
	return fma(fma(Float64(y * y), Float64(x * fma(t_0, 0.16666666666666666, 0.5)), 1.0), t_0, 1.0)
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(y * y), $MachinePrecision] * N[(x * N[(t$95$0 * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0 + 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

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 71.9%

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

   1. *-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 72.3

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

7. Applied egg-rr 72.3%

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

Alternative 13: 71.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 71.9%

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. associate-+l+ N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

7. Applied egg-rr 72.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 72.2%

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

11. Final simplification 72.2%

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

Alternative 14: 70.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0\right)\right), 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (fma
  (* x (* y y))
  (* x (* x (fma 0.16666666666666666 (* (* y y) (* y y)) 0.0)))
  1.0))

C

double code(double x, double y) {
	return fma((x * (y * y)), (x * (x * fma(0.16666666666666666, ((y * y) * (y * y)), 0.0))), 1.0);
}

Julia

function code(x, y)
	return fma(Float64(x * Float64(y * y)), Float64(x * Float64(x * fma(0.16666666666666666, Float64(Float64(y * y) * Float64(y * y)), 0.0))), 1.0)
end

Wolfram

code[x_, y_] := N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{6}\right) + \frac{1}{2} \cdot {y}^{4}\right) + {y}^{2}\right)} \]

5. Simplified 71.9%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{\frac{1}{6} \cdot \left({x}^{2} \cdot {y}^{4}\right)}, 1\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{\left({x}^{2} \cdot {y}^{4}\right) \cdot \frac{1}{6}}, 1\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \left(\color{blue}{\left(x \cdot x\right)} \cdot {y}^{4}\right) \cdot \frac{1}{6}, 1\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{\left(x \cdot \left(x \cdot {y}^{4}\right)\right)} \cdot \frac{1}{6}, 1\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(x \cdot {y}^{4}\right)\right)}, 1\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{4}\right)\right)}, 1\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(\left(x \cdot {y}^{4}\right) \cdot \frac{1}{6}\right)}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(x \cdot \left({y}^{4} \cdot \frac{1}{6}\right)\right)}, 1\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right), 1\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} \cdot {y}^{4}\right)\right)}, 1\right) \]

   11. +-rgt-identity N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{4} + 0\right)}\right), 1\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{4}, 0\right)}\right), 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{\color{blue}{\left(2 \cdot 2\right)}}, 0\right)\right), 1\right) \]

   14. pow-sqr N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot {y}^{2}}, 0\right)\right), 1\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{{y}^{2} \cdot {y}^{2}}, 0\right)\right), 1\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}, 0\right)\right), 1\right) \]

   17. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(y \cdot y\right)} \cdot {y}^{2}, 0\right)\right), 1\right) \]

   18. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{6}, \left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}, 0\right)\right), 1\right) \]

   19. *-lowering-*.f64 71.9

       \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), x \cdot \left(x \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)}, 0\right)\right), 1\right) \]

8. Simplified 71.9%

   \[\leadsto \mathsf{fma}\left(x \cdot \left(y \cdot y\right), \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot \left(y \cdot y\right), 0\right)\right)}, 1\right) \]
9. Add Preprocessing

Alternative 15: 70.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot y\right) \cdot \mathsf{fma}\left(x \cdot y, y \cdot 0.5, 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e+88)
   (fma (* (* x y) (fma (* x y) (* y 0.5) 1.0)) y 1.0)
   (* x (* 0.5 (* x (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e+88) {
		tmp = fma(((x * y) * fma((x * y), (y * 0.5), 1.0)), y, 1.0);
	} else {
		tmp = x * (0.5 * (x * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e+88)
		tmp = fma(Float64(Float64(x * y) * fma(Float64(x * y), Float64(y * 0.5), 1.0)), y, 1.0);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e+88], N[(N[(N[(x * y), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * N[(y * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(0.5 * N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.64999999999999994e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 73.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. associate-*r* N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 73.9

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

   7. Applied egg-rr 73.9%

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

   if 2.64999999999999994e88 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right) + 1} \]

   5. Simplified 53.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot {y}^{4}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left({y}^{4} \cdot {x}^{2}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot {x}^{2}} \]

      3. unpow2 N/A

         \[\leadsto \left(\frac{1}{2} \cdot {y}^{4}\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{2} \cdot {y}^{4}\right) \cdot x\right)} \]

      7. associate-*r* N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({y}^{4} \cdot x\right)\right)} \]

      8. *-commutative N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right)\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot {y}^{4}\right)}\right) \]

      11. metadata-eval N/A

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

      12. pow-sqr N/A

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

      13. *-lowering-*.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 59.0

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

   8. Simplified 59.0%

      \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot y\right) \cdot \mathsf{fma}\left(x \cdot y, y \cdot 0.5, 1\right), y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e+27) (fma (* x y) y 1.0) (* 0.5 (* x (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e+27) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = 0.5 * (x * (x * (y * y)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e+27)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = Float64(0.5 * Float64(x * Float64(x * Float64(y * y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e+27], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(0.5 * N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e27

   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 64.6

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

   5. Simplified 64.6%

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

      1. associate-*r* N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 64.7

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

   7. Applied egg-rr 64.7%

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

   if 1e27 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 40.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. *-commutative N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 42.8

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

   7. Simplified 42.8%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 84.5

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

   10. Simplified 84.5%

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

   11. Taylor expanded in x around inf

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

       1. *-lowering-*.f64 N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 84.5

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

   13. Simplified 84.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.5) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	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)) <= 0.5d0) then
        tmp = 1.0d0
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.5) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 0.5:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 0.5)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.5

   1. Initial program 100.0%

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

   4. Applied egg-rr 67.4%

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

   if 0.5 < (*.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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 66.9

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

   5. Simplified 66.9%

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

   6. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 66.9

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

   8. Simplified 66.9%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y (* x y)) 5e+47) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 5e+47) {
		tmp = 1.0;
	} else {
		tmp = 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 ((y * (x * y)) <= 5d+47) then
        tmp = 1.0d0
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 5e+47) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 5e+47:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 5e+47)
		tmp = 1.0;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 5e+47)
		tmp = 1.0;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 5e+47], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 5.00000000000000022e47

   1. Initial program 100.0%

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

   4. Applied egg-rr 64.3%

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

   if 5.00000000000000022e47 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied egg-rr 41.3%

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

   5. Taylor expanded in x 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. accelerator-lowering-fma.f64 17.0

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

   7. Simplified 17.0%

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

   8. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 16.9

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

   10. Simplified 16.9%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 67.3

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

5. Simplified 67.3%

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

Alternative 20: 51.5% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 47.1%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))