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

Percentage Accurate: 100.0% → 100.0%

Time: 32.9s

Alternatives: 18

Speedup: 1.0×

• 24.8% 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 4.0) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 4.0) {
		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 (exp((y * (x * y))) <= 4.0d0) 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 (Math.exp((y * (x * y))) <= 4.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp((y * (x * y))) <= 4.0:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 4.0)
		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 (exp((y * (x * y))) <= 4.0)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4.0], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 4

   1. Initial program 99.9%

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

   4. Applied egg-rr 60.5%

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

   if 4 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 99.8%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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 60.2

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

   5. Simplified 60.2%

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

   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 60.2

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

   8. Simplified 60.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 4:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400.0)
   (exp x)
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400.0) {
		tmp = exp(x);
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400.0)
		tmp = exp(x);
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400.0], N[Exp[x], $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   if -400 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.4e-81)
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)
   (exp (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-81) {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	} else {
		tmp = exp((x * y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-81)
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	else
		tmp = exp(Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.4e-81], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.39999999999999979e-81

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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 73.1%

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

   if 5.39999999999999979e-81 < y

   1. Initial program 99.9%

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

   4. Applied egg-rr 89.1%

      \[\leadsto e^{\color{blue}{x} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -400.0)
     (* 0.16666666666666666 (* x (* x x)))
     (if (<= t_0 1.0)
       (fma x (* y y) 1.0)
       (if (<= t_0 2e+78)
         (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
         (fma x (fma x (* (* y y) 0.5) y) 1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = fma(x, (y * y), 1.0);
	} else if (t_0 <= 2e+78) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = fma(x, fma(x, ((y * y) * 0.5), y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 1.0)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (t_0 <= 2e+78)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 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_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+78], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 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 4 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y) < 1

   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 99.1

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

   5. Simplified 99.1%

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

   if 1 < (*.f64 (*.f64 x y) y) < 2.00000000000000002e78

   1. Initial program 99.2%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 41.0

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

   7. Simplified 41.0%

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

   if 2.00000000000000002e78 < (*.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 47.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 93.8

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

   7. Simplified 93.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -400.0)
     (* 0.16666666666666666 (* x (* x x)))
     (if (<= t_0 1.0)
       (fma x (* y y) 1.0)
       (if (<= t_0 2e+78)
         (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
         (* x (* x (* (* y y) 0.5))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = fma(x, (y * y), 1.0);
	} else if (t_0 <= 2e+78) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = x * (x * ((y * y) * 0.5));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 1.0)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (t_0 <= 2e+78)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(Float64(y * y) * 0.5)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+78], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y) < 1

   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 99.1

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

   5. Simplified 99.1%

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

   if 1 < (*.f64 (*.f64 x y) y) < 2.00000000000000002e78

   1. Initial program 99.2%

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

   4. Applied egg-rr 55.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 41.0

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

   7. Simplified 41.0%

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

   if 2.00000000000000002e78 < (*.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 47.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 93.8

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

   7. Simplified 93.8%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 93.8

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

   10. Simplified 93.8%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.5 \cdot \left(y \cdot y\right)\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -400.0)
     (* 0.16666666666666666 (* x (* x x)))
     (if (<= t_0 1.0)
       (fma x (* y y) 1.0)
       (if (<= t_0 5e+160)
         (fma x (fma x (fma x 0.16666666666666666 0.5) 1.0) 1.0)
         (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = fma(x, (y * y), 1.0);
	} else if (t_0 <= 5e+160) {
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 1.0)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (t_0 <= 5e+160)
		tmp = fma(x, fma(x, fma(x, 0.16666666666666666, 0.5), 1.0), 1.0);
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+160], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y) < 1

   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 99.1

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

   5. Simplified 99.1%

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

   if 1 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e160

   1. Initial program 99.5%

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

   4. Applied egg-rr 61.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 47.2

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

   7. Simplified 47.2%

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

   if 5.0000000000000002e160 < (*.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 88.4

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

   5. Simplified 88.4%

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

   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 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right), 1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -400.0)
     (* 0.16666666666666666 (* x (* x x)))
     (if (<= t_0 1.0)
       (fma x (* y y) 1.0)
       (if (<= t_0 5e+160)
         (* x (* x (fma x 0.16666666666666666 0.5)))
         (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 1.0) {
		tmp = fma(x, (y * y), 1.0);
	} else if (t_0 <= 5e+160) {
		tmp = x * (x * fma(x, 0.16666666666666666, 0.5));
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 1.0)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (t_0 <= 5e+160)
		tmp = Float64(x * Float64(x * fma(x, 0.16666666666666666, 0.5)));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+160], N[(x * N[(x * N[(x * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y) < 1

   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 99.1

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

   5. Simplified 99.1%

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

   if 1 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e160

   1. Initial program 99.5%

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

   4. Applied egg-rr 61.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 47.2

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

   7. Simplified 47.2%

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

   8. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. associate-*l* N/A

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

      9. lft-mult-inverse N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. accelerator-lowering-fma.f64 46.7

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

   10. Simplified 46.7%

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

   if 5.0000000000000002e160 < (*.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 88.4

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

   5. Simplified 88.4%

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

   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 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+160}:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, 0.16666666666666666, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400.0)
   (* 0.16666666666666666 (* x (* x x)))
   (fma
    (* y y)
    (fma (* x (* x (* y y))) (fma x (* (* y y) 0.16666666666666666) 0.5) x)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else {
		tmp = fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = fma(Float64(y * y), fma(Float64(x * Float64(x * Float64(y * y))), fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 92.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right), \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ t_1 := 0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+160}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))) (t_1 (* 0.16666666666666666 (* x (* x x)))))
   (if (<= t_0 -400.0)
     t_1
     (if (<= t_0 1.0)
       (fma x (* y y) 1.0)
       (if (<= t_0 5e+160) t_1 (* x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double t_1 = 0.16666666666666666 * (x * (x * x));
	double tmp;
	if (t_0 <= -400.0) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = fma(x, (y * y), 1.0);
	} else if (t_0 <= 5e+160) {
		tmp = t_1;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	t_1 = Float64(0.16666666666666666 * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = fma(x, Float64(y * y), 1.0);
	elseif (t_0 <= 5e+160)
		tmp = t_1;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], t$95$1, If[LessEqual[t$95$0, 1.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+160], t$95$1, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -400 or 1 < (*.f64 (*.f64 x y) y) < 5.0000000000000002e160

   1. Initial program 99.8%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 11.4

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

   7. Simplified 11.4%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 24.3

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

   10. Simplified 24.3%

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

   if -400 < (*.f64 (*.f64 x y) y) < 1

   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 99.1

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

   5. Simplified 99.1%

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

   if 5.0000000000000002e160 < (*.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 88.4

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

   5. Simplified 88.4%

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

   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 88.4

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

   8. Simplified 88.4%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 1:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{+160}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 -400.0)
     (* 0.16666666666666666 (* x (* x x)))
     (if (<= t_0 500.0)
       (fma x (* y y) 1.0)
       (* x (* x (* 0.5 (* (* y y) (* y y)))))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else if (t_0 <= 500.0) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = x * (x * (0.5 * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	elseif (t_0 <= 500.0)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 * Float64(Float64(y * y) * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * N[(0.5 * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y) < 500

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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 98.3

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

   5. Simplified 98.3%

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

   if 500 < (*.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 75.7%

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

   6. Taylor expanded in y 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. *-commutative N/A

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

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

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

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

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

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 80.7

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

   8. Simplified 80.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 500:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x, \left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5, x\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400.0)
   (* 0.16666666666666666 (* x (* x x)))
   (fma (* y y) (fma x (* (* x (* y y)) 0.5) x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else {
		tmp = fma((y * y), fma(x, ((x * (y * y)) * 0.5), x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = fma(Float64(y * y), fma(x, Float64(Float64(x * Float64(y * y)) * 0.5), x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 91.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(x, \left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5, x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400.0)
   (* 0.16666666666666666 (* x (* x x)))
   (fma (* y y) (* x (* (* x (* y y)) 0.5)) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -400.0) {
		tmp = 0.16666666666666666 * (x * (x * x));
	} else {
		tmp = fma((y * y), (x * ((x * (y * y)) * 0.5)), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -400.0)
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	else
		tmp = fma(Float64(y * y), Float64(x * Float64(Float64(x * Float64(y * y)) * 0.5)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400.0], N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(x * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -400

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 2.2

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

   7. Simplified 2.2%

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

   8. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{3}} \]

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 18.6

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

   10. Simplified 18.6%

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

   if -400 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 91.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 90.4

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

   8. Simplified 90.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, x \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot 0.5\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 5e-67) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 5e-67) {
		tmp = 1.0;
	} else {
		tmp = fma(x, y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 5e-67)
		tmp = 1.0;
	else
		tmp = fma(x, y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 5e-67], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 4.9999999999999999e-67

   1. Initial program 99.9%

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

   4. Applied egg-rr 60.2%

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

   if 4.9999999999999999e-67 < (*.f64 (*.f64 x y) y)

   1. Initial program 99.8%

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

   4. Applied egg-rr 46.7%

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

   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 15.1

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

   7. Simplified 15.1%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 5 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1.0) {
		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)) <= 1.0d0) 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)) <= 1.0) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1.0)
		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)) <= 1.0)
		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], 1.0], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   4. Applied egg-rr 60.5%

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

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

   1. Initial program 99.8%

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

   4. Applied egg-rr 44.5%

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

   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 10.8

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

   7. Simplified 10.8%

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

   8. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 10.6

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

   10. Simplified 10.6%

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

4. Final simplification 48.9%

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

Alternative 16: 65.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(x, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.5e+159) (fma x (* y y) 1.0) (fma x (fma x 0.5 1.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.5e+159) {
		tmp = fma(x, (y * y), 1.0);
	} else {
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.5e+159)
		tmp = fma(x, Float64(y * y), 1.0);
	else
		tmp = fma(x, fma(x, 0.5, 1.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.5e+159], N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.5000000000000003e159

   1. Initial program 99.9%

      \[e^{\left(x \cdot y\right) \cdot y} \]
   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 60.9

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

   5. Simplified 60.9%

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

   if 9.5000000000000003e159 < x

   1. Initial program 99.6%

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

   4. Applied egg-rr 83.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 83.2

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

   7. Simplified 83.2%

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

Alternative 17: 66.7% 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 99.9%

   \[e^{\left(x \cdot y\right) \cdot y} \]
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 60.4

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

5. Simplified 60.4%

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

Alternative 18: 51.4% 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 99.9%

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

4. Applied egg-rr 47.1%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))