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

Percentage Accurate: 100.0% → 100.0%

Time: 30.4s

Alternatives: 20

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 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 82.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y (* x y))) 0.0)
   (exp (* x y))
   (fma
    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 (exp((y * (x * y))) <= 0.0) {
		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 (exp(Float64(y * Float64(x * y))) <= 0.0)
		tmp = exp(Float64(x * y));
	else
		tmp = fma(x, Float64(Float64(y * y) * Float64(Float64(x * y) * Float64(y * fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5)))), fma(Float64(x * y), y, 1.0));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 43.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.7%

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 93.7%

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

4. Final simplification 78.1%

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

Alternative 3: 70.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 57.7

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 57.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 83.9%

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

   12. Applied rewrites 86.8%

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

4. Final simplification 65.1%

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

Alternative 4: 70.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 57.7

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

   5. Applied rewrites 57.7%

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

   7. Applied rewrites 57.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 83.9%

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

4. Final simplification 64.4%

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

Alternative 5: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right)\right), \mathsf{fma}\left(x \cdot y, y, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) -400000.0)
   (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)) <= -400000.0) {
		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)) <= -400000.0)
		tmp = exp(x);
	else
		tmp = fma(x, Float64(Float64(y * y) * Float64(Float64(x * y) * Float64(y * fma(x, Float64(Float64(y * y) * 0.16666666666666666), 0.5)))), fma(Float64(x * y), y, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], -400000.0], N[Exp[x], $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < -4e5

   1. Initial program 100.0%

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

   4. Applied rewrites 67.3%

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

   if -4e5 < (*.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 93.7%

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

   7. Applied rewrites 93.7%

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

   9. Applied rewrites 93.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq -400000:\\ \;\;\;\;e^{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \left(\left(x \cdot y\right) \cdot \left(y \cdot \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.16666666666666666, 0.5\right)\right)\right), \mathsf{fma}\left(x \cdot y, y, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+144}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 2e+15)
     (fma (* x y) y 1.0)
     (if (<= t_0 1e+144)
       (* (* y y) (* y (* 0.16666666666666666 (* x (* x x)))))
       (* (* x (* x (* y y))) (* (* y y) 0.5))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (t_0 <= 2e+15) {
		tmp = fma((x * y), y, 1.0);
	} else if (t_0 <= 1e+144) {
		tmp = (y * y) * (y * (0.16666666666666666 * (x * (x * x))));
	} else {
		tmp = (x * (x * (y * y))) * ((y * y) * 0.5);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (t_0 <= 2e+15)
		tmp = fma(Float64(x * y), y, 1.0);
	elseif (t_0 <= 1e+144)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(0.16666666666666666 * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(y * y))) * 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, 2e+15], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+144], N[(N[(y * y), $MachinePrecision] * N[(y * N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.4%

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

   if 2e15 < (*.f64 (*.f64 x y) y) < 1.00000000000000002e144

   1. Initial program 100.0%

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

   4. Applied rewrites 51.1%

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

   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 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 24.6

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

   10. Applied rewrites 24.6%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 36.6%

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

   if 1.00000000000000002e144 < (*.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 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 10^{+144}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot \left(y \cdot y\right)\right)\right) \cdot \left(\left(y \cdot y\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+144}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= t_0 2e+15)
     (fma (* x y) y 1.0)
     (if (<= t_0 1e+144)
       (* (* y y) (* y (* 0.16666666666666666 (* x (* x x)))))
       (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 <= 2e+15) {
		tmp = fma((x * y), y, 1.0);
	} else if (t_0 <= 1e+144) {
		tmp = (y * y) * (y * (0.16666666666666666 * (x * (x * x))));
	} 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 <= 2e+15)
		tmp = fma(Float64(x * y), y, 1.0);
	elseif (t_0 <= 1e+144)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(0.16666666666666666 * Float64(x * Float64(x * x)))));
	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, 2e+15], N[(N[(x * y), $MachinePrecision] * y + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 1e+144], N[(N[(y * y), $MachinePrecision] * N[(y * N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.4%

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

   if 2e15 < (*.f64 (*.f64 x y) y) < 1.00000000000000002e144

   1. Initial program 100.0%

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

   4. Applied rewrites 51.1%

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

   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 2.9

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

   7. Applied rewrites 2.9%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r* N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 24.6

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

   10. Applied rewrites 24.6%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 36.6%

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

   if 1.00000000000000002e144 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 52.8%

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

   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

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      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 94.4

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

   7. Applied rewrites 94.4%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, y, 1\right)\\ \mathbf{elif}\;y \cdot \left(x \cdot y\right) \leq 10^{+144}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot 0.5, y\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot {y}^{4}\right) + {y}^{2}\right)} \]
   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 57.6%

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

   if 2e15 < (*.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 86.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 91.1%

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

4. Final simplification 66.0%

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

Alternative 9: 71.6% accurate, 1.9× speedup

Math

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

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

Wolfram

code[x_, y_] := N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(x * y), $MachinePrecision] * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] * y + 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 65.0%

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

7. Applied rewrites 65.0%

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

9. Applied rewrites 65.0%

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

10. Final simplification 65.0%

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

Alternative 10: 71.6% accurate, 1.9× speedup

Math

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

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 * y) * Float64(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 * y), $MachinePrecision] * N[(y * N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $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 65.0%

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

7. Applied rewrites 65.0%

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

8. Final simplification 65.0%

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

Alternative 11: 71.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 65.0%

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

7. Applied rewrites 65.0%

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

Alternative 12: 71.4% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (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) {
	return fma((y * y), fma((x * (x * (y * y))), fma(x, ((y * y) * 0.16666666666666666), 0.5), x), 1.0);
}

Julia

function code(x, y)
	return 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

Wolfram

code[x_, y_] := 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. 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 65.0%

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

Alternative 13: 71.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := 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]), $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

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

5. Applied rewrites 65.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 64.6%

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

Alternative 14: 68.4% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* 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 ((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 (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[(y * N[(x * y), $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 (*.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 57.4

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

   5. Applied rewrites 57.4%

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

   7. Applied rewrites 57.4%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 52.4%

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

   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

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      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 77.8

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

   7. Applied rewrites 77.8%

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

4. Final simplification 62.5%

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

Alternative 15: 70.0% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] * N[(x * N[(0.5 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $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 63.7%

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

6. Final simplification 63.7%

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

Alternative 16: 65.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.1) 1.0 (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.1) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * (x * y)) <= 0.1d0) then
        tmp = 1.0d0
    else
        tmp = x * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.1) {
		tmp = 1.0;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 0.1:
		tmp = 1.0
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 0.1)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (x * y)) <= 0.1)
		tmp = 1.0;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision], 0.1], 1.0, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.5%

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

   if 0.10000000000000001 < (*.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 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 53.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* x y)) 0.1) 1.0 (fma x y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (x * y)) <= 0.1) {
		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)) <= 0.1)
		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], 0.1], 1.0, N[(x * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 0.10000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.5%

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

   if 0.10000000000000001 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 51.7%

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

   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. lower-fma.f64 14.7

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

   7. Applied rewrites 14.7%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(x \cdot y\right) \leq 0.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y):
	tmp = 0
	if (y * (x * y)) <= 2e+15:
		tmp = 1.0
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(x * y)) <= 2e+15)
		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)) <= 2e+15)
		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], 2e+15], 1.0, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.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

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

   5. Applied rewrites 57.3%

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

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

   1. Initial program 100.0%

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

   4. Applied rewrites 52.4%

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

   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 14.9

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 14.7%

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

4. Final simplification 46.6%

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

Alternative 19: 65.8% 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 59.1

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

5. Applied rewrites 59.1%

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

Alternative 20: 50.6% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))