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

Percentage Accurate: 100.0% → 100.0%

Time: 32.5s

Alternatives: 16

Speedup: 1.0×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* (* x y) y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp(((x * y) * y))
end function

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* (* x y) y)))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp(((x * y) * y))
end function

Java

public static double code(double x, double y) {
	return Math.exp(((x * y) * y));
}

Python

def code(x, y):
	return math.exp(((x * y) * y))

Julia

function code(x, y)
	return exp(Float64(Float64(x * y) * y))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (exp (* y (* x y))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((y * (x * y)))
end function

Java

public static double code(double x, double y) {
	return Math.exp((y * (x * y)));
}

Python

def code(x, y):
	return math.exp((y * (x * y)))

Julia

function code(x, y)
	return exp(Float64(y * Float64(x * y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 73.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 2e15

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.6%

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

   if 2e15 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 88.8

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

   12. Applied rewrites 88.8%

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

4. Final simplification 73.2%

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

Alternative 3: 73.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 67.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 84.9%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*r* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

       16. lower-*.f64 87.6

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

   12. Applied rewrites 87.6%

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

4. Final simplification 73.2%

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

Alternative 4: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot \left(x \cdot y\right)} \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 2e+15)
   (fma (* x y) y 1.0)
   (fma x (fma x (* (* y y) 0.5) y) 1.0)))

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * Float64(x * y))) <= 2e+15)
		tmp = fma(Float64(x * y), y, 1.0);
	else
		tmp = fma(x, fma(x, Float64(Float64(y * y) * 0.5), y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e+15], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 (*.f64 (*.f64 x y) y)) < 2e15

   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. lower-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. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 67.3

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

   7. Applied rewrites 67.3%

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

   if 2e15 < (exp.f64 (*.f64 (*.f64 x y) y))

   1. Initial program 100.0%

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

   4. Applied rewrites 38.1%

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

   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. lower-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. lower-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. lower-*.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. lower-*.f64 77.5

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

   7. Applied rewrites 77.5%

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

4. Final simplification 70.0%

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

Alternative 5: 67.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (exp((y * (x * y))) <= 2.0) {
		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))) <= 2.0d0) 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))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.exp((y * (x * y))) <= 2.0:
		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))) <= 2.0)
		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))) <= 2.0)
		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], 2.0], 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)) < 2

   1. Initial program 100.0%

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

   4. Applied rewrites 67.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-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. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.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. lower-*.f64 65.6

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

   8. Applied rewrites 65.6%

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

4. Final simplification 66.7%

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

Alternative 6: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -1000000:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right), \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -1000000.0)
   (exp (* x y))
   (fma
    x
    (* (* y y) (* (* x (* y y)) (fma x (* (* y y) 0.16666666666666666) 0.5)))
    (fma x (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -1000000.0) {
		tmp = exp((x * y));
	} else {
		tmp = fma(x, ((y * y) * ((x * (y * y)) * fma(x, ((y * y) * 0.16666666666666666), 0.5))), fma(x, (y * y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -1000000.0)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(x, Float64(Float64(y * y) * Float64(Float64(x * Float64(y * y)) * fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5))), fma(x, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -1000000.0], N[Exp[N[(x * y), $MachinePrecision]], $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -1e6

   1. Initial program 100.0%

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

   4. Applied rewrites 47.0%

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

   if -1e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

   7. Applied rewrites 94.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -1000000:\\ \;\;\;\;e^{x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right), \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -1000000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right), \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -1000000.0)
   (exp x)
   (fma
    x
    (* (* y y) (* (* x (* y y)) (fma x (* (* y y) 0.16666666666666666) 0.5)))
    (fma x (* y y) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= -1000000.0) {
		tmp = exp(x);
	} else {
		tmp = fma(x, ((y * y) * ((x * (y * y)) * fma(x, ((y * y) * 0.16666666666666666), 0.5))), fma(x, (y * y), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= -1000000.0)
		tmp = exp(x);
	else
		tmp = fma(x, Float64(Float64(y * y) * Float64(Float64(x * Float64(y * y)) * fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5))), fma(x, Float64(y * y), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -1000000.0], N[Exp[x], $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -1e6

   1. Initial program 100.0%

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

   4. Applied rewrites 63.5%

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

   if -1e6 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+l+ N/A

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

   7. Applied rewrites 94.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -1000000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right), \mathsf{fma}\left(x, y \cdot y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

   10. associate-+l+ N/A

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

7. Applied rewrites 72.5%

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

8. Final simplification 72.5%

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

Alternative 9: 71.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(0.5 \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e+20)
   (fma (* x y) y 1.0)
   (* x (* x (* 0.5 (* (* y y) (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e+20) {
		tmp = fma((x * y), y, 1.0);
	} else {
		tmp = x * (x * (0.5 * ((y * y) * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e+20)
		tmp = fma(Float64(x * 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_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 1e+20], N[(N[(x * y), $MachinePrecision] * y + 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 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1e20

   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. lower-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. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if 1e20 < (*.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. Applied rewrites 86.6%

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

   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. lower-*.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. lower-*.f64 N/A

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

      9. lower-*.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. lower-*.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. lower-*.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. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

4. Final simplification 72.6%

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

Alternative 10: 72.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-*.f64 N/A

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

   10. associate-+l+ N/A

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

7. Applied rewrites 72.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 71.5%

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

11. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f64 N/A

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

    5. *-commutative N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. lower-*.f64 N/A

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

    9. lower-*.f64 71.5

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

13. Applied rewrites 71.5%

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

14. Final simplification 71.5%

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

Alternative 11: 71.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := 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. 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. Applied rewrites 71.3%

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

Alternative 12: 70.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
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. Applied rewrites 71.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. associate-+l+ N/A

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

7. Applied rewrites 70.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-+r+ N/A

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

9. Applied rewrites 71.2%

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

Alternative 13: 54.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 1e-12) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-12) {
		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)) <= 1e-12)
		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], 1e-12], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   4. Applied rewrites 67.1%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 37.6%

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

   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. lower-fma.f64 12.2

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

   7. Applied rewrites 12.2%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y (* x y)) 1e-12) 1.0 (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 1e-12) {
		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)) <= 1d-12) 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)) <= 1e-12) {
		tmp = 1.0;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 1e-12:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 1e-12)
		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)) <= 1e-12)
		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], 1e-12], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   4. Applied rewrites 67.1%

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

   if 9.9999999999999998e-13 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 37.6%

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

   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. lower-fma.f64 12.2

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

   7. Applied rewrites 12.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 11.9

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

   10. Applied rewrites 11.9%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 10^{-12}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.0% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-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. lower-*.f64 67.1

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

5. Applied rewrites 67.1%

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

Alternative 16: 51.3% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 49.9%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))