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

Percentage Accurate: 100.0% → 100.0%

Time: 20.0s

Alternatives: 12

Speedup: 1.0×

• 24.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 65.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (exp(((x * y) * y)) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if math.exp(((x * y) * y)) <= 2.0:
		tmp = 1.0
	else:
		tmp = (y * y) * x
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (exp(((x * y) * y)) <= 2.0)
		tmp = 1.0;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

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

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

   5. Applied rewrites 65.1%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 61.6

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

   5. Applied rewrites 61.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 61.6%

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

Alternative 3: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot x\\ t_1 := \left(x \cdot y\right) \cdot y\\ \mathbf{if}\;t\_1 \leq -20000:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t\_0 \cdot x, 0.5, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(t\_0, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x, y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) x)) (t_1 (* (* x y) y)))
   (if (<= t_1 -20000.0)
     (exp x)
     (if (<= t_1 2000.0)
       (fma (fma (* t_0 x) 0.5 x) (* y y) 1.0)
       (fma
        (fma (* (* (fma t_0 0.16666666666666666 0.5) x) x) (* y y) x)
        (* y y)
        1.0)))))

C

double code(double x, double y) {
	double t_0 = (y * y) * x;
	double t_1 = (x * y) * y;
	double tmp;
	if (t_1 <= -20000.0) {
		tmp = exp(x);
	} else if (t_1 <= 2000.0) {
		tmp = fma(fma((t_0 * x), 0.5, x), (y * y), 1.0);
	} else {
		tmp = fma(fma(((fma(t_0, 0.16666666666666666, 0.5) * x) * x), (y * y), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * x)
	t_1 = Float64(Float64(x * y) * y)
	tmp = 0.0
	if (t_1 <= -20000.0)
		tmp = exp(x);
	elseif (t_1 <= 2000.0)
		tmp = fma(fma(Float64(t_0 * x), 0.5, x), Float64(y * y), 1.0);
	else
		tmp = fma(fma(Float64(Float64(fma(t_0, 0.16666666666666666, 0.5) * x) * x), Float64(y * y), x), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$1, -20000.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(N[(t$95$0 * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 * 0.16666666666666666 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < -2e4

   1. Initial program 100.0%

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

   4. Applied rewrites 56.5%

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

   if -2e4 < (*.f64 (*.f64 x y) y) < 2e3

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.9

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

   5. Applied rewrites 98.9%

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 89.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.0%

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

   if 2e3 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   7. Applied rewrites 44.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 74.2%

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

   12. Applied rewrites 89.0%

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

Alternative 4: 71.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 10^{-65}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.16666666666666666, 0.5\right) \cdot x\right) \cdot x, y \cdot y, x\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* x y) y) 1e-65)
   1.0
   (fma
    (fma (* (* (fma (* (* y y) x) 0.16666666666666666 0.5) x) x) (* y y) x)
    (* y y)
    1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 1e-65) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(((fma(((y * y) * x), 0.16666666666666666, 0.5) * x) * x), (y * y), x), (y * y), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 1e-65)
		tmp = 1.0;
	else
		tmp = fma(fma(Float64(Float64(fma(Float64(Float64(y * y) * x), 0.16666666666666666, 0.5) * x) * x), Float64(y * y), x), Float64(y * y), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 1e-65], 1.0, N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 9.99999999999999923e-66

   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 64.3%

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

   if 9.99999999999999923e-66 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 75.1%

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

   12. Applied rewrites 88.6%

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

Alternative 5: 61.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-123}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+123}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.16666666666666666, \left(y \cdot y\right) \cdot x, 0.5\right), y \cdot y, x\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot y\right) \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-123)
   1.0
   (if (<= y 1.75e+123)
     (fma
      (fma (* (* x x) (fma 0.16666666666666666 (* (* y y) x) 0.5)) (* y y) x)
      (* y y)
      1.0)
     (fma (* (* 0.16666666666666666 y) y) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-123) {
		tmp = 1.0;
	} else if (y <= 1.75e+123) {
		tmp = fma(fma(((x * x) * fma(0.16666666666666666, ((y * y) * x), 0.5)), (y * y), x), (y * y), 1.0);
	} else {
		tmp = fma(((0.16666666666666666 * y) * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-123)
		tmp = 1.0;
	elseif (y <= 1.75e+123)
		tmp = fma(fma(Float64(Float64(x * x) * fma(0.16666666666666666, Float64(Float64(y * y) * x), 0.5)), Float64(y * y), x), Float64(y * y), 1.0);
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * y) * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-123], 1.0, If[LessEqual[y, 1.75e+123], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.7000000000000001e-123

   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 60.9%

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

   if 2.7000000000000001e-123 < y < 1.75e123

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   10. Applied rewrites 63.1%

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

   if 1.75e123 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 49.5

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

   7. Applied rewrites 49.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 49.5%

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

Alternative 6: 65.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x y) y)))
   (if (<= t_0 2e+29) 1.0 (if (<= t_0 4e+304) (* (* 0.5 y) y) (* (* y x) y)))))

C

double code(double x, double y) {
	double t_0 = (x * y) * y;
	double tmp;
	if (t_0 <= 2e+29) {
		tmp = 1.0;
	} else if (t_0 <= 4e+304) {
		tmp = (0.5 * y) * y;
	} else {
		tmp = (y * x) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * y) * y
    if (t_0 <= 2d+29) then
        tmp = 1.0d0
    else if (t_0 <= 4d+304) then
        tmp = (0.5d0 * y) * y
    else
        tmp = (y * x) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * y) * y;
	double tmp;
	if (t_0 <= 2e+29) {
		tmp = 1.0;
	} else if (t_0 <= 4e+304) {
		tmp = (0.5 * y) * y;
	} else {
		tmp = (y * x) * y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * y) * y
	tmp = 0
	if t_0 <= 2e+29:
		tmp = 1.0
	elif t_0 <= 4e+304:
		tmp = (0.5 * y) * y
	else:
		tmp = (y * x) * y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * y) * y)
	tmp = 0.0
	if (t_0 <= 2e+29)
		tmp = 1.0;
	elseif (t_0 <= 4e+304)
		tmp = Float64(Float64(0.5 * y) * y);
	else
		tmp = Float64(Float64(y * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * y) * y;
	tmp = 0.0;
	if (t_0 <= 2e+29)
		tmp = 1.0;
	elseif (t_0 <= 4e+304)
		tmp = (0.5 * y) * y;
	else
		tmp = (y * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+29], 1.0, If[LessEqual[t$95$0, 4e+304], N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(y * x), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x y) y) < 1.99999999999999983e29

   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 63.8%

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

   if 1.99999999999999983e29 < (*.f64 (*.f64 x y) y) < 3.9999999999999998e304

   1. Initial program 100.0%

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 39.9

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

   7. Applied rewrites 39.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 39.8%

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

   if 3.9999999999999998e304 < (*.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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

Alternative 7: 70.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 64.1

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

5. Applied rewrites 64.1%

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

7. Applied rewrites 59.7%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

10. Applied rewrites 62.3%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 66.7%

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

Alternative 8: 60.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-54}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot y\right) \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-54)
   1.0
   (if (<= y 8.2e+106)
     (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)
     (fma (* (* 0.16666666666666666 y) y) y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-54) {
		tmp = 1.0;
	} else if (y <= 8.2e+106) {
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = fma(((0.16666666666666666 * y) * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-54)
		tmp = 1.0;
	elseif (y <= 8.2e+106)
		tmp = fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * y) * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-54], 1.0, If[LessEqual[y, 8.2e+106], N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.50000000000000008e-54

   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 61.7%

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

   if 2.50000000000000008e-54 < y < 8.2000000000000005e106

   1. Initial program 100.0%

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 47.3

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

   7. Applied rewrites 47.3%

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

   if 8.2000000000000005e106 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 49.6

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 49.6%

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

Alternative 9: 62.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x \cdot y\right) \cdot y \leq 2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (* x y) y) 2e+29) 1.0 (* (* 0.5 y) y)))

C

double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 2e+29) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * y) * y) <= 2d+29) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * y) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * y) * y) <= 2e+29) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * y) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * y) * y) <= 2e+29:
		tmp = 1.0
	else:
		tmp = (0.5 * y) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * y) * y) <= 2e+29)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * y) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * y) * y) <= 2e+29)
		tmp = 1.0;
	else
		tmp = (0.5 * y) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] * y), $MachinePrecision], 2e+29], 1.0, N[(N[(0.5 * y), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x y) y) < 1.99999999999999983e29

   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 63.8%

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

   if 1.99999999999999983e29 < (*.f64 (*.f64 x y) y)

   1. Initial program 100.0%

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 57.6

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 57.6%

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

Alternative 10: 64.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.16666666666666666 \cdot y\right) \cdot y, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.05e+116)
   (fma (* y x) y 1.0)
   (fma (* (* 0.16666666666666666 y) y) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.05e+116) {
		tmp = fma((y * x), y, 1.0);
	} else {
		tmp = fma(((0.16666666666666666 * y) * y), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.05e+116)
		tmp = fma(Float64(y * x), y, 1.0);
	else
		tmp = fma(Float64(Float64(0.16666666666666666 * y) * y), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.05e+116], N[(N[(y * x), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.05000000000000009e116

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   7. Applied rewrites 65.4%

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

   if 3.05000000000000009e116 < y

   1. Initial program 100.0%

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 49.5

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

   7. Applied rewrites 49.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 49.5%

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

Alternative 11: 65.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 64.1

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

5. Applied rewrites 64.1%

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

Alternative 12: 50.9% 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 50.1%

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

Reproduce

herbie shell --seed 2024324 
(FPCore (x y)
  :name "Data.Random.Distribution.Normal:normalF from random-fu-0.2.6.2"
  :precision binary64
  (exp (* (* x y) y)))