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

Percentage Accurate: 100.0% → 100.0%

Time: 32.3s

Alternatives: 18

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 81.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   4. Applied rewrites 97.6%

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

      1. *-rgt-identity 97.6

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

   6. Applied rewrites 97.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 69.9

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

   8. Applied rewrites 69.9%

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

Alternative 3: 56.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 97.6%

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

      1. *-rgt-identity 97.6

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

   6. Applied rewrites 97.6%

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

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

   1. Initial program 100.0%

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

      1. *-rgt-identity N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-out N/A

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

      6. div-inv N/A

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

      7. div-inv N/A

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

      8. flip-+ N/A

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

      9. +-inverses N/A

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

      10. +-inverses N/A

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

      11. associate-*r/ N/A

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

      12. *-rgt-identity N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out N/A

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

      17. div-inv N/A

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

      18. div-inv N/A

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

      19. +-inverses N/A

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

      20. difference-of-squares N/A

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

      21. +-inverses N/A

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

      22. flip-+ N/A

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

      23. count-2 N/A

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

   4. Applied rewrites 49.2%

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

   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. lower-fma.f64 14.7

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 14.7

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

   10. Applied rewrites 14.7%

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

Alternative 4: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

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

   11. associate-*r/ N/A

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

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

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

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. 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. Add Preprocessing

Alternative 5: 92.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
4. 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. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. lift-*.f64 N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lower-fma.f64 N/A

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

7. Applied rewrites 94.1%

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

8. Final simplification 94.1%

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

Alternative 6: 92.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
4. 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. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

Alternative 7: 93.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

Alternative 8: 91.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
4. 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. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 93.8

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

8. Applied rewrites 93.8%

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

Alternative 9: 92.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
4. 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. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(y \cdot y, \color{blue}{{y}^{4} \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2} \cdot \frac{x}{{y}^{2}}\right)}, x\right) \]
7. Step-by-step derivation

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. pow-sqr N/A

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

   10. associate-*l* N/A

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

   11. associate-*r* N/A

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

   12. associate-/l* N/A

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

   13. *-rgt-identity N/A

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

   14. associate-*r/ N/A

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

   15. rgt-mult-inverse N/A

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

   16. *-rgt-identity N/A

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

   17. *-commutative N/A

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

8. Applied rewrites 93.5%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 93.5%

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

Alternative 10: 90.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 1 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. pow-plus N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow-plus N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. cube-mult N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 86.1

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

   8. Applied rewrites 86.1%

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

Alternative 11: 92.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + {y}^{2} \cdot \left(\frac{1}{6} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{2} \cdot x\right)\right)} \]
4. 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. lower-fma.f64 N/A

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

5. Applied rewrites 94.1%

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

6. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 93.5

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

8. Applied rewrites 93.5%

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

Alternative 12: 91.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 92.5

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

5. Applied rewrites 92.5%

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

Alternative 13: 89.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
4. 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. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 91.4%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{fma}\left(y, x \cdot \color{blue}{\left(\frac{1}{2} \cdot {y}^{3}\right)}, x\right) \]
7. Step-by-step derivation

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 91.0

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

8. Applied rewrites 91.0%

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

Alternative 14: 67.6% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

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

   11. associate-*r/ N/A

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

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

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

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. 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. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 70.0

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

7. Applied rewrites 70.0%

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

Alternative 15: 82.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 84.4

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

5. Applied rewrites 84.4%

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

Alternative 16: 56.1% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

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

   11. associate-*r/ N/A

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

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

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

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. 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

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

   1. +-commutative N/A

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

   2. lower-+.f64 56.6

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

7. Applied rewrites 56.6%

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

Alternative 17: 56.1% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-rgt-identity N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-out N/A

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

   6. div-inv N/A

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

   7. div-inv N/A

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

   8. flip-+ N/A

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

   9. +-inverses N/A

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

   10. +-inverses N/A

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

   11. associate-*r/ N/A

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

   12. *-rgt-identity N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-lft-out N/A

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

   17. div-inv N/A

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

   18. div-inv N/A

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

   19. +-inverses N/A

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

   20. difference-of-squares N/A

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

   21. +-inverses N/A

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

   22. flip-+ N/A

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

   23. 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. lower-fma.f64 56.6

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

7. Applied rewrites 56.6%

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

Alternative 18: 51.4% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 52.2%

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

   1. *-rgt-identity 52.2

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

6. Applied rewrites 52.2%

   \[\leadsto \color{blue}{x} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024214 
(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))))