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

Percentage Accurate: 100.0% → 100.0%

Time: 34.2s

Alternatives: 17

Speedup: 1.0×

• 50.0% 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: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{y \cdot y} \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right), y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (exp (* y y)) 2.0)
   (* x (fma y y 1.0))
   (* x (fma (* y y) (fma y 0.16666666666666666 0.5) y))))

C

double code(double x, double y) {
	double tmp;
	if (exp((y * y)) <= 2.0) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * fma((y * y), fma(y, 0.16666666666666666, 0.5), y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (exp(Float64(y * y)) <= 2.0)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * fma(Float64(y * y), fma(y, 0.16666666666666666, 0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision] + 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

   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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

      1. *-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

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 52.1%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 42.9

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

   7. Simplified 42.9%

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

   8. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-lft-identity N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. lft-mult-inverse N/A

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

      19. metadata-eval N/A

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

      20. accelerator-lowering-fma.f64 42.9

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

   10. Simplified 42.9%

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

Alternative 3: 74.3% 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

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.8%

   \[\leadsto x \cdot e^{\color{blue}{y}} \]
5. Add Preprocessing

Alternative 4: 94.1% 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. accelerator-lowering-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. Simplified 93.4%

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

Alternative 5: 90.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-8)
   (* x (fma y y 1.0))
   (* x (* y (fma y (* (* y y) 0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * fma(y, ((y * y) * 0.5), y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * fma(y, Float64(Float64(y * y) * 0.5), y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-plus N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. unpow3 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lft-mult-inverse N/A

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

      12. *-lft-identity N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

   8. Simplified 84.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot 0.5, y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $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. accelerator-lowering-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. Simplified 93.4%

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

6. Taylor expanded in y around inf

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

   1. *-lowering-*.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 93.4

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

8. Simplified 93.4%

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

9. Final simplification 93.4%

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

Alternative 7: 90.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e-8) (* x (fma y y 1.0)) (* x (* y (* 0.5 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * (0.5 * (y * (y * y))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(0.5 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   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. accelerator-lowering-fma.f64 100.0

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

   5. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)} \]
   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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 84.1

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

   5. Simplified 84.1%

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

   6. Taylor expanded in y around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 84.0

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

   8. Simplified 84.0%

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

Alternative 8: 68.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(y \cdot \mathsf{fma}\left(y, 0.16666666666666666, 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5.0)
   (* x (fma y y 1.0))
   (* x (* y (* y (fma y 0.16666666666666666 0.5))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5.0) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * (y * fma(y, 0.16666666666666666, 0.5)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5.0)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(y * fma(y, 0.16666666666666666, 0.5))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5.0], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(y * N[(y * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2}\right)} \]
   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. accelerator-lowering-fma.f64 99.3

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

   5. Simplified 99.3%

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

   if 5 < (*.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

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 52.5%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 43.2

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

   7. Simplified 43.2%

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

   8. Taylor expanded in y around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lft-mult-inverse N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 43.2

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

   10. Simplified 43.2%

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

Alternative 9: 68.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 20:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 20.0)
   (* x (fma y y 1.0))
   (* x (* y (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 20.0) {
		tmp = x * fma(y, y, 1.0);
	} else {
		tmp = x * (y * ((y * y) * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 20.0)
		tmp = Float64(x * fma(y, y, 1.0));
	else
		tmp = Float64(x * Float64(y * Float64(Float64(y * y) * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 20.0], N[(x * N[(y * y + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 20

   1. Initial program 99.9%

      \[x \cdot e^{y \cdot y} \]
   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. accelerator-lowering-fma.f64 98.7

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

   5. Simplified 98.7%

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

   if 20 < (*.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

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 52.8%

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

   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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 43.5

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

   7. Simplified 43.5%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 43.5

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

   10. Simplified 43.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 20:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.f64 92.2

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

5. Simplified 92.2%

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

Alternative 11: 68.0% 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

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.8%

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

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. accelerator-lowering-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. accelerator-lowering-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. accelerator-lowering-fma.f64 71.3

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

7. Simplified 71.3%

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

Alternative 12: 80.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-8) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		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 ((y * y) <= 5d-8) 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 ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-8:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = x;
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-8)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2}\right)} \]
   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. accelerator-lowering-fma.f64 69.5

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

   5. Simplified 69.5%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.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. *-lowering-*.f64 69.5

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

   8. Simplified 69.5%

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

Alternative 13: 56.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 5e-8) x (* x y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e-8) {
		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 ((y * y) <= 5d-8) 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 ((y * y) <= 5e-8) {
		tmp = x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e-8:
		tmp = x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e-8)
		tmp = x;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e-8)
		tmp = x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e-8], x, N[(x * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.9999999999999998e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   if 4.9999999999999998e-8 < (*.f64 y y)

   1. Initial program 99.9%

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

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

      24. associate-*r/ N/A

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

   4. Applied egg-rr 52.1%

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

   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. accelerator-lowering-fma.f64 16.5

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

   7. Simplified 16.5%

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

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 16.5

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

   10. Simplified 16.5%

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

Alternative 14: 81.1% accurate, 9.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2}\right)} \]
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. accelerator-lowering-fma.f64 85.1

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

5. Simplified 85.1%

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

Alternative 15: 56.0% 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

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.8%

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

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. +-lowering-+.f64 58.4

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

7. Simplified 58.4%

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

Alternative 16: 56.0% 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

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

   24. associate-*r/ N/A

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

4. Applied egg-rr 75.8%

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

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. accelerator-lowering-fma.f64 58.4

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

7. Simplified 58.4%

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

Alternative 17: 51.1% 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

3. Taylor expanded in y around 0

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

5. Simplified 52.8%

   \[\leadsto \color{blue}{x} \]
6. 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 2024198 
(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))))