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

Percentage Accurate: 100.0% → 100.0%

Time: 7.8s

Alternatives: 15

Speedup: 1.0×

• 50.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 68.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y y)) 2.0)
   (fma (* y x) y x)
   (* (* (* (fma 0.16666666666666666 y 0.5) y) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * y)) <= 2.0) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = ((fma(0.16666666666666666, y, 0.5) * y) * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * y)) <= 2.0)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(Float64(fma(0.16666666666666666, y, 0.5) * y) * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(N[(0.16666666666666666 * y + 0.5), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   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. 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 56.4%

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

   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 41.7

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 41.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 41.7%

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

4. Final simplification 67.0%

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

Alternative 3: 68.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y y)) 2.0)
   (fma (* y x) y x)
   (* (* (* 0.16666666666666666 y) (* y y)) x)))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * y)) <= 2.0) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = ((0.16666666666666666 * y) * (y * y)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * y)) <= 2.0)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(Float64(0.16666666666666666 * y) * Float64(y * y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + x \cdot {y}^{2}} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

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

   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. 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 56.4%

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

   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 41.7

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 41.7%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 41.7%

       \[\leadsto x \cdot \left(\left(0.16666666666666666 \cdot y\right) \cdot \left(y \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

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

Alternative 4: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot y} \cdot x \leq 5 \cdot 10^{+304}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (exp (* y y)) x) 5e+304) (* 1.0 x) (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((exp((y * y)) * x) <= 5e+304) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((exp((y * y)) * x) <= 5d+304) then
        tmp = 1.0d0 * x
    else
        tmp = (y * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.exp((y * y)) * x) <= 5e+304) {
		tmp = 1.0 * x;
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.exp((y * y)) * x) <= 5e+304:
		tmp = 1.0 * x
	else:
		tmp = (y * y) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(exp(Float64(y * y)) * x) <= 5e+304)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((exp((y * y)) * x) <= 5e+304)
		tmp = 1.0 * x;
	else
		tmp = (y * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], 5e+304], N[(1.0 * x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (*.f64 y y))) < 4.9999999999999997e304

   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 61.7%

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

   if 4.9999999999999997e304 < (*.f64 x (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

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot y} \cdot x \leq 5 \cdot 10^{+304}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.5)
   (fma
    (* (* (* y x) y) (* y y))
    (fma (* y y) 0.16666666666666666 0.5)
    (fma (* y y) x x))
   (* (exp y) x)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 5.4%

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

   9. Taylor expanded in y around 0

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

       1. distribute-lft-in N/A

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

       2. *-commutative N/A

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

       3. associate-+r+ N/A

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

       4. +-commutative N/A

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

       5. distribute-lft-in N/A

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

       6. distribute-lft-in N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. +-commutative N/A

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

   11. Applied rewrites 99.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), \mathsf{fma}\left(y \cdot y, x, x\right)\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

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

   if 0.5 < (*.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 56.4%

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

4. Final simplification 75.5%

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

Alternative 6: 56.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(1.0 * x), $MachinePrecision], N[(y * 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 97.4%

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

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

   1. Initial program 100.0%

      \[x \cdot e^{y \cdot y} \]
   2. Add Preprocessing
   3. 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 56.4%

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

   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 14.6

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

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 14.6%

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

4. Final simplification 51.5%

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

Alternative 7: 71.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.5, x\right), y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, y, -0.5\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e+153)
   (fma (fma (* (* y y) x) 0.5 x) (* y y) x)
   (*
    (/
     (* (fma 0.027777777777777776 (* y y) -0.25) (* y y))
     (fma 0.16666666666666666 y -0.5))
    x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e+153) {
		tmp = fma(fma(((y * y) * x), 0.5, x), (y * y), x);
	} else {
		tmp = ((fma(0.027777777777777776, (y * y), -0.25) * (y * y)) / fma(0.16666666666666666, y, -0.5)) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e+153)
		tmp = fma(fma(Float64(Float64(y * y) * x), 0.5, x), Float64(y * y), x);
	else
		tmp = Float64(Float64(Float64(fma(0.027777777777777776, Float64(y * y), -0.25) * Float64(y * y)) / fma(0.16666666666666666, y, -0.5)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e+153], N[(N[(N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision] * 0.5 + x), $MachinePrecision] * N[(y * y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(0.027777777777777776 * N[(y * y), $MachinePrecision] + -0.25), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 * y + -0.5), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 5.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
   10. 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

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if 2e153 < (*.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 55.7%

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

   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 50.8

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

   7. Applied rewrites 50.8%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 50.8%

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

   12. Applied rewrites 55.0%

       \[\leadsto x \cdot \frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{y}, -0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, 0.5, x\right), y \cdot y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.027777777777777776, y \cdot y, -0.25\right) \cdot \left(y \cdot y\right)}{\mathsf{fma}\left(0.16666666666666666, y, -0.5\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 37.7%

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

9. Taylor expanded in y around 0

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

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. associate-+r+ N/A

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

    4. +-commutative N/A

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

    5. distribute-lft-in N/A

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

    6. distribute-lft-in N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 92.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), \mathsf{fma}\left(y \cdot y, x, x\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 92.0%

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

Alternative 9: 91.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 37.7%

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

9. Taylor expanded in y around 0

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

    1. distribute-lft-in N/A

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

    2. *-commutative N/A

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

    3. associate-+r+ N/A

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

    4. +-commutative N/A

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

    5. distribute-lft-in N/A

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

    6. distribute-lft-in N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. +-commutative N/A

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

11. Applied rewrites 92.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{4} \cdot x, \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 0.5\right), \mathsf{fma}\left(y \cdot y, x, x\right)\right)} \]
12. Step-by-step derivation

13. Applied rewrites 91.6%

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

Alternative 10: 87.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 79.2

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

5. Applied rewrites 79.2%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 37.7%

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

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + {y}^{2} \cdot \left(x + \frac{1}{2} \cdot \left(x \cdot {y}^{2}\right)\right)} \]
10. 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

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 88.1

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

11. Applied rewrites 88.1%

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

Alternative 11: 81.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 1e+127) (fma (* y x) y x) (* (* y y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1e+127) {
		tmp = fma((y * x), y, x);
	} else {
		tmp = (y * y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1e+127)
		tmp = fma(Float64(y * x), y, x);
	else
		tmp = Float64(Float64(y * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1e+127], N[(N[(y * x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.99999999999999955e126

   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 81.7

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

   5. Applied rewrites 81.7%

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

   7. Applied rewrites 81.7%

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

   if 9.99999999999999955e126 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. lower-fma.f64 76.3

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. associate-*r/ N/A

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

   13. *-rgt-identity N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. distribute-lft-out N/A

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

   18. div-inv N/A

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

   19. div-inv N/A

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

   20. +-inverses N/A

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

   21. difference-of-squares N/A

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

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.0%

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

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 65.8

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

7. Applied rewrites 65.8%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 66.5%

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

11. Final simplification 66.5%

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

Alternative 13: 81.6% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 79.2

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

5. Applied rewrites 79.2%

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

Alternative 14: 55.5% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. associate-*r/ N/A

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

   13. *-rgt-identity N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. distribute-lft-out N/A

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

   18. div-inv N/A

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

   19. div-inv N/A

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

   20. +-inverses N/A

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

   21. difference-of-squares N/A

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

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.0%

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

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 50.8

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

7. Applied rewrites 50.8%

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

Alternative 15: 9.6% accurate, 18.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-out N/A

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

   7. div-inv N/A

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

   8. div-inv N/A

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

   9. flip-+ N/A

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

   10. +-inverses N/A

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

   11. +-inverses N/A

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

   12. associate-*r/ N/A

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

   13. *-rgt-identity N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. distribute-lft-out N/A

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

   18. div-inv N/A

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

   19. div-inv N/A

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

   20. +-inverses N/A

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

   21. difference-of-squares N/A

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

   22. +-inverses N/A

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

   23. flip-+ N/A

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

   24. count-2 N/A

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

4. Applied rewrites 74.0%

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

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 50.8

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

7. Applied rewrites 50.8%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 10.3%

    \[\leadsto y \cdot \color{blue}{x} \]
11. 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 2024259 
(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))))