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

Percentage Accurate: 100.0% → 100.0%

Time: 22.4s

Alternatives: 20

Speedup: 1.0×

• 49.7% 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, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ {\left({\left(e^{y\_m}\right)}^{\left(0.5 \cdot y\_m\right)}\right)}^{2} \cdot x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (pow (pow (exp y_m) (* 0.5 y_m)) 2.0) x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return pow(pow(exp(y_m), (0.5 * y_m)), 2.0) * x;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = ((exp(y_m) ** (0.5d0 * y_m)) ** 2.0d0) * x
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return Math.pow(Math.pow(Math.exp(y_m), (0.5 * y_m)), 2.0) * x;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return math.pow(math.pow(math.exp(y_m), (0.5 * y_m)), 2.0) * x

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(((exp(y_m) ^ Float64(0.5 * y_m)) ^ 2.0) * x)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((exp(y_m) ^ (0.5 * y_m)) ^ 2.0) * x;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[Power[N[Power[N[Exp[y$95$m], $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 100.0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

   \[\leadsto x \cdot {\left(\sqrt{{\left(e^{y}\right)}^{y}}\right)}^{\color{blue}{2}} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

   \[\leadsto x \cdot {\left({\left(e^{y}\right)}^{\left(0.5 \cdot y\right)}\right)}^{2} \]

10. Final simplification 100.0%

    \[\leadsto {\left({\left(e^{y}\right)}^{\left(0.5 \cdot y\right)}\right)}^{2} \cdot x \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* (pow (exp y_m) y_m) x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[Power[N[Exp[y$95$m], $MachinePrecision], y$95$m], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 75.8% accurate, 0.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (exp (* y_m y_m)) 2.0) (* 1.0 x) (* (* y_m x) y_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[Exp[N[(y$95$m * y$95$m), $MachinePrecision]], $MachinePrecision], 2.0], N[(1.0 * x), $MachinePrecision], N[(N[(y$95$m * x), $MachinePrecision] * y$95$m), $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 99.9%

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

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

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.9%

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

   10. Applied rewrites 44.6%

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

4. Final simplification 71.4%

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

Alternative 4: 63.7% accurate, 0.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (exp (* y_m y_m)) 2.0) (* 1.0 x) (* y_m x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[Exp[N[(y$95$m * y$95$m), $MachinePrecision]], $MachinePrecision], 2.0], N[(1.0 * x), $MachinePrecision], N[(y$95$m * 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 99.9%

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

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

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 48.7%

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

   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 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 12.5%

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

4. Final simplification 54.8%

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

Alternative 5: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ e^{y\_m \cdot y\_m} \cdot x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* (exp (* y_m y_m)) x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[Exp[N[(y$95$m * y$95$m), $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 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ {\left(y\_m - -1\right)}^{y\_m} \cdot x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* (pow (- y_m -1.0) y_m) x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return pow((y_m - -1.0), y_m) * x;
}

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return Math.pow((y_m - -1.0), y_m) * x;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return math.pow((y_m - -1.0), y_m) * x

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64((Float64(y_m - -1.0) ^ y_m) * x)
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((y_m - -1.0) ^ y_m) * x;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[Power[N[(y$95$m - -1.0), $MachinePrecision], y$95$m], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in y around 0

   \[\leadsto x \cdot {\left(1 + y\right)}^{y} \]
7. Step-by-step derivation

8. Applied rewrites 73.4%

   \[\leadsto x \cdot {\left(y - -1\right)}^{y} \]

9. Final simplification 73.4%

   \[\leadsto {\left(y - -1\right)}^{y} \cdot x \]
10. Add Preprocessing

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ e^{y\_m} \cdot x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* (exp y_m) x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[Exp[y$95$m], $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 73.3%

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

5. Final simplification 73.3%

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

Alternative 8: 89.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y\_m \cdot y\_m, 0.5, 1\right) \cdot x\right) \cdot y\_m, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y\_m, 0.5\right), y\_m, 1\right), y\_m, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 2e+204)
   (fma (* (* (fma (* y_m y_m) 0.5 1.0) x) y_m) y_m x)
   (* (fma (fma (fma 0.16666666666666666 y_m 0.5) y_m 1.0) y_m 1.0) x)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 2e+204) {
		tmp = fma(((fma((y_m * y_m), 0.5, 1.0) * x) * y_m), y_m, x);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, y_m, 0.5), y_m, 1.0), y_m, 1.0) * x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 2e+204)
		tmp = fma(Float64(Float64(fma(Float64(y_m * y_m), 0.5, 1.0) * x) * y_m), y_m, x);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, y_m, 0.5), y_m, 1.0), y_m, 1.0) * x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 2e+204], N[(N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * y$95$m + 0.5), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999998e204

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   12. Applied rewrites 82.3%

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

   if 1.99999999999999998e204 < (*.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 49.6%

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

   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 48.8

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

   7. Applied rewrites 48.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(y \cdot y, 0.5, 1\right) \cdot x\right) \cdot y, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.2% accurate, 2.8× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma
  (fma (* (fma 0.16666666666666666 (* y_m y_m) 0.5) x) (* y_m y_m) x)
  (* y_m y_m)
  x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(fma((fma(0.16666666666666666, (y_m * y_m), 0.5) * x), (y_m * y_m), x), (y_m * y_m), x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(fma(Float64(fma(0.16666666666666666, Float64(y_m * y_m), 0.5) * x), Float64(y_m * y_m), x), Float64(y_m * y_m), x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(N[(0.16666666666666666 * N[(y$95$m * y$95$m), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + x), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around inf

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

   1. unpow2 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 100.0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

10. Applied rewrites 91.5%

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

11. Final simplification 91.5%

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

Alternative 10: 89.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.5 \cdot y\_m\right) \cdot x\right) \cdot y\_m, y\_m \cdot y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y\_m, 0.5\right), y\_m, 1\right), y\_m, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 2e+204)
   (fma (* (* (* 0.5 y_m) x) y_m) (* y_m y_m) x)
   (* (fma (fma (fma 0.16666666666666666 y_m 0.5) y_m 1.0) y_m 1.0) x)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 2e+204) {
		tmp = fma((((0.5 * y_m) * x) * y_m), (y_m * y_m), x);
	} else {
		tmp = fma(fma(fma(0.16666666666666666, y_m, 0.5), y_m, 1.0), y_m, 1.0) * x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 2e+204)
		tmp = fma(Float64(Float64(Float64(0.5 * y_m) * x) * y_m), Float64(y_m * y_m), x);
	else
		tmp = Float64(fma(fma(fma(0.16666666666666666, y_m, 0.5), y_m, 1.0), y_m, 1.0) * x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 2e+204], N[(N[(N[(N[(0.5 * y$95$m), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * y$95$m + 0.5), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.99999999999999998e204

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt1-in N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 82.2%

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

   if 1.99999999999999998e204 < (*.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 49.6%

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

   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 48.8

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

   7. Applied rewrites 48.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+204}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(0.5 \cdot y\right) \cdot x\right) \cdot y, y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, y, 0.5\right), y, 1\right), y, 1\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot x, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(0.16666666666666666, y\_m, 0.5\right) \cdot y\_m\right) \cdot x\right) \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 5.0)
   (fma (* y_m x) y_m x)
   (* (* (* (fma 0.16666666666666666 y_m 0.5) y_m) x) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 5.0) {
		tmp = fma((y_m * x), y_m, x);
	} else {
		tmp = ((fma(0.16666666666666666, y_m, 0.5) * y_m) * x) * y_m;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 5.0)
		tmp = fma(Float64(y_m * x), y_m, x);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, y_m, 0.5) * y_m) * x) * y_m);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 5.0], N[(N[(y$95$m * x), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * y$95$m + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision] * x), $MachinePrecision] * y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5

   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 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   if 5 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 48.9%

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

   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 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 30.8

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

   10. Applied rewrites 30.8%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 32.2%

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

Alternative 12: 91.1% accurate, 4.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma (* (fma (* y_m y_m) 0.5 1.0) (* y_m y_m)) x x))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(y$95$m * y$95$m), $MachinePrecision]), $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 inf

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

   1. unpow2 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 100.0

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 86.5

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

10. Applied rewrites 86.5%

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

12. Applied rewrites 89.7%

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

Alternative 13: 87.4% accurate, 4.6× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (fma (fma (fma 0.16666666666666666 y_m 0.5) y_m 1.0) y_m 1.0) x))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(N[(0.16666666666666666 * y$95$m + 0.5), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision] * y$95$m + 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 73.3%

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

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 66.9

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

7. Applied rewrites 66.9%

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

8. Final simplification 66.9%

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

Alternative 14: 84.3% accurate, 4.6× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma (fma (* (fma 0.16666666666666666 y_m 0.5) y_m) x x) y_m x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma(fma((fma(0.16666666666666666, y_m, 0.5) * y_m), x, x), y_m, x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(fma(Float64(fma(0.16666666666666666, y_m, 0.5) * y_m), x, x), y_m, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(N[(0.16666666666666666 * y$95$m + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision] * x + x), $MachinePrecision] * y$95$m + 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 73.3%

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

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 54.6

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 64.0

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

10. Applied rewrites 64.0%

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

12. Applied rewrites 64.7%

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

Alternative 15: 82.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(y\_m \cdot x, y\_m, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot y\_m\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 5e+48) (fma (* y_m x) y_m x) (* (* y_m y_m) x)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 5e+48) {
		tmp = fma((y_m * x), y_m, x);
	} else {
		tmp = (y_m * y_m) * x;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 5e+48)
		tmp = fma(Float64(y_m * x), y_m, x);
	else
		tmp = Float64(Float64(y_m * y_m) * x);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 5e+48], N[(N[(y$95$m * x), $MachinePrecision] * y$95$m + x), $MachinePrecision], N[(N[(y$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.99999999999999973e48

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 92.3

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

   5. Applied rewrites 92.3%

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

   7. Applied rewrites 92.3%

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

   if 4.99999999999999973e48 < (*.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 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.5%

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

Alternative 16: 81.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \cdot y\_m \leq 4 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot y\_m\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* y_m y_m) 4e-15) (* 1.0 x) (* (* y_m y_m) x)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 4e-15) {
		tmp = 1.0 * x;
	} else {
		tmp = (y_m * y_m) * x;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m * y_m) <= 4d-15) then
        tmp = 1.0d0 * x
    else
        tmp = (y_m * y_m) * x
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m * y_m) <= 4e-15) {
		tmp = 1.0 * x;
	} else {
		tmp = (y_m * y_m) * x;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m * y_m) <= 4e-15:
		tmp = 1.0 * x
	else:
		tmp = (y_m * y_m) * x
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(y_m * y_m) <= 4e-15)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y_m * y_m) * x);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m * y_m) <= 4e-15)
		tmp = 1.0 * x;
	else
		tmp = (y_m * y_m) * x;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(y$95$m * y$95$m), $MachinePrecision], 4e-15], N[(1.0 * x), $MachinePrecision], N[(N[(y$95$m * y$95$m), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.0000000000000003e-15

   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 99.9%

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

   if 4.0000000000000003e-15 < (*.f64 y y)

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 58.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4 \cdot 10^{-15}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.9% accurate, 5.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (fma (* (* (* y_m x) y_m) 0.16666666666666666) y_m x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return fma((((y_m * x) * y_m) * 0.16666666666666666), y_m, x);
}

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(Float64(Float64(Float64(y_m * x) * y_m) * 0.16666666666666666), y_m, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(N[(y$95$m * x), $MachinePrecision] * y$95$m), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y$95$m + 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 73.3%

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

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 54.6

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 64.0

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

10. Applied rewrites 64.0%

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

11. Taylor expanded in y around inf

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

13. Applied rewrites 64.3%

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

14. Final simplification 64.3%

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

Alternative 18: 82.0% accurate, 9.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (fma (* y_m y_m) x x))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(y$95$m * y$95$m), $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 78.8

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

5. Applied rewrites 78.8%

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

Alternative 19: 63.1% accurate, 15.9× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (fma y_m x x))

C

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

Julia

y_m = abs(y)
function code(x, y_m)
	return fma(y_m, x, x)
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(y$95$m * 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 73.3%

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

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 54.6

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

7. Applied rewrites 54.6%

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

Alternative 20: 17.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y\_m \cdot x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* y_m x))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return y_m * x;
}

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return y_m * x

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(y_m * x)
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(y$95$m * 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 73.3%

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

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 54.6

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 8.5%

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

11. Final simplification 8.5%

    \[\leadsto y \cdot x \]
12. 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 2024296 
(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))))