Data.Number.Erf:$dmerfcx from erf-2.0.0.0

Percentage Accurate: 100.0% → 100.0%

Time: 21.5s

Alternatives: 15

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{y \cdot y} \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto e^{y \cdot y} \cdot x \]
4. Add Preprocessing

Alternative 2: 81.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.1%

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

4. Final simplification 79.2%

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

Alternative 3: 75.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.5%

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

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

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 59.1%

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

   10. Applied rewrites 46.8%

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

4. Final simplification 73.0%

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

Alternative 4: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{y} \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   11. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   12. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   13. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   16. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   17. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   19. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   20. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   21. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.6%

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

5. Final simplification 74.6%

   \[\leadsto e^{y} \cdot x \]
6. Add Preprocessing

Alternative 5: 72.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.5, 1\right), \left(y \cdot y\right) \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot \left(y \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+206)
   (fma (fma (* y y) 0.5 1.0) (* (* y y) x) x)
   (* (* (fma 0.16666666666666666 y 0.5) (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+206) {
		tmp = fma(fma((y * y), 0.5, 1.0), ((y * y) * x), x);
	} else {
		tmp = (fma(0.16666666666666666, y, 0.5) * (y * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+206)
		tmp = fma(fma(Float64(y * y), 0.5, 1.0), Float64(Float64(y * y) * x), x);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, y, 0.5) * Float64(y * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+206], N[(N[(N[(y * y), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1e206

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 87.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(0.16666666666666666, y \cdot y, 0.5\right), \mathsf{fma}\left(y, y, 1\right) \cdot x\right)} \]

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. associate-+r+ N/A

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

       6. distribute-lft-in N/A

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

       9. +-commutative N/A

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

   11. Applied rewrites 87.0%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 82.1%

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

   if 1e206 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out N/A

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

      7. div-inv N/A

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

      8. div-inv N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

      11. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      12. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      13. *-rgt-identity N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

      14. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

      15. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

      17. distribute-lft-out N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

      19. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      20. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

      21. difference-of-squares N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

      24. count-2 N/A

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 52.5

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

   7. Applied rewrites 52.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 52.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+206}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.5, 1\right), \left(y \cdot y\right) \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot \left(y \cdot y\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(0.16666666666666666, y \cdot y, 0.5\right), \mathsf{fma}\left(y, y, 1\right) \cdot x\right)} \]

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-+r+ N/A

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

    6. distribute-lft-in N/A

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

    7. associate-+r+ N/A

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

    8. +-commutative N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 91.0%

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

13. Applied rewrites 93.3%

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

Alternative 7: 94.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(0.16666666666666666, y \cdot y, 0.5\right), \mathsf{fma}\left(y, y, 1\right) \cdot x\right)} \]

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-+r+ N/A

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

    6. distribute-lft-in N/A

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

    7. associate-+r+ N/A

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

    8. +-commutative N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 91.0%

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

13. Applied rewrites 93.2%

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

Alternative 8: 92.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 91.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(0.16666666666666666, y \cdot y, 0.5\right), \mathsf{fma}\left(y, y, 1\right) \cdot x\right)} \]

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. distribute-lft-in N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. associate-+r+ N/A

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

    6. distribute-lft-in N/A

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

    7. associate-+r+ N/A

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

    8. +-commutative N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 91.0%

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

12. Taylor expanded in y around inf

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

14. Applied rewrites 91.0%

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

Alternative 9: 68.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y \cdot y, y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-6)
   (fma (* y x) y x)
   (* (fma (fma 0.16666666666666666 y 0.5) (* y y) y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-6) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = fma(fma(0.16666666666666666, y, 0.5), (y * y), y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-6)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(fma(fma(0.16666666666666666, y, 0.5), Float64(y * y), y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-6], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999991e-6

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   if 1.99999999999999991e-6 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out N/A

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

      7. div-inv N/A

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

      8. div-inv N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

      11. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      12. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      13. *-rgt-identity N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

      14. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

      15. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

      17. distribute-lft-out N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

      19. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      20. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

      21. difference-of-squares N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

      24. count-2 N/A

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 37.1

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 37.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y \cdot y, y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot \left(y \cdot y\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-6)
   (fma (* y x) y x)
   (* (* (fma 0.16666666666666666 y 0.5) (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-6) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = (fma(0.16666666666666666, y, 0.5) * (y * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-6)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(fma(0.16666666666666666, y, 0.5) * Float64(y * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-6], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999991e-6

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   if 1.99999999999999991e-6 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-out N/A

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

      7. div-inv N/A

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

      8. div-inv N/A

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

      9. flip-+ N/A

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

      10. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

      11. +-inverses N/A

          \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

      12. associate-*r/ N/A

          \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

      13. *-rgt-identity N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

      14. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

      15. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

      16. metadata-eval N/A

          \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

      17. distribute-lft-out N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

      18. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

      19. div-inv N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

      20. +-inverses N/A

          \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

      21. difference-of-squares N/A

          \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

      22. +-inverses N/A

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

      23. flip-+ N/A

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

      24. count-2 N/A

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

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 37.1

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

   7. Applied rewrites 37.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 37.1%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right) \cdot \left(y \cdot y\right)\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+90) (fma (* y x) y x) (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+90) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+90)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+90], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999993e90

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   7. Applied rewrites 88.0%

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

   if 1.99999999999999993e90 < (*.f64 y y)

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 67.8%

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

Alternative 12: 67.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   11. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   12. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   13. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   16. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   17. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   19. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   20. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   21. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.6%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 67.5

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

7. Applied rewrites 67.5%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 67.5%

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

11. Final simplification 67.5%

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

Alternative 13: 81.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 79.3

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

5. Applied rewrites 79.3%

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

Alternative 14: 55.6% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	return fma(y, x, x)
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{\color{blue}{0}}{\frac{y}{2} - \frac{y}{2}}} \]

   11. +-inverses N/A

       \[\leadsto x \cdot e^{y \cdot \frac{0}{\color{blue}{0}}} \]

   12. associate-*r/ N/A

       \[\leadsto x \cdot e^{\color{blue}{\frac{y \cdot 0}{0}}} \]

   13. *-rgt-identity N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot 1\right)} \cdot 0}{0}} \]

   14. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) \cdot 0}{0}} \]

   15. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\color{blue}{\frac{1}{2}} + \frac{1}{2}\right)\right) \cdot 0}{0}} \]

   16. metadata-eval N/A

       \[\leadsto x \cdot e^{\frac{\left(y \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right)\right) \cdot 0}{0}} \]

   17. distribute-lft-out N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)} \cdot 0}{0}} \]

   18. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\color{blue}{\frac{y}{2}} + y \cdot \frac{1}{2}\right) \cdot 0}{0}} \]

   19. div-inv N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \color{blue}{\frac{y}{2}}\right) \cdot 0}{0}} \]

   20. +-inverses N/A

       \[\leadsto x \cdot e^{\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \color{blue}{\left(\frac{y}{2} - \frac{y}{2}\right)}}{0}} \]

   21. difference-of-squares N/A

       \[\leadsto x \cdot e^{\frac{\color{blue}{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}}{0}} \]

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.6%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 54.8

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

7. Applied rewrites 54.8%

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

Alternative 15: 50.9% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 * x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

5. Applied rewrites 51.2%

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

6. Final simplification 51.2%

   \[\leadsto 1 \cdot x \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (pow (exp y) y)))

  (* x (exp (* y y))))