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

Percentage Accurate: 100.0% → 100.0%

Time: 12.0s

Alternatives: 12

Speedup: 1.0×

• 50.2% 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: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 40:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \left(0.5 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 40.0)
   (*
    x
    (+
     1.0
     (* (* y y) (+ 1.0 (* y (* y (+ 0.5 (* (* y y) 0.16666666666666666))))))))
   (* x (exp y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 40.0) {
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))));
	} else {
		tmp = x * exp(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) <= 40.0d0) then
        tmp = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * (0.5d0 + ((y * y) * 0.16666666666666666d0)))))))
    else
        tmp = x * exp(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 40.0) {
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))));
	} else {
		tmp = x * Math.exp(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 40.0:
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))))
	else:
		tmp = x * math.exp(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 40.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(1.0 + Float64(y * Float64(y * Float64(0.5 + Float64(Float64(y * y) * 0.16666666666666666))))))));
	else
		tmp = Float64(x * exp(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 40.0)
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))));
	else
		tmp = x * exp(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 40.0], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * N[(0.5 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 40

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 99.3%

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

   if 40 < (*.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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot 1\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right)\right)\right)\right) \]

      5. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + y \cdot \frac{1}{2}\right)\right)\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \left(\frac{y}{2} + \frac{y}{2}\right)\right)\right)\right) \]

      8. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      9. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      10. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(y \cdot \frac{0}{0}\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y \cdot 0}{0}\right)\right)\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot 1\right) \cdot 0}{0}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \left(\frac{1}{2} + \frac{1}{2}\right)\right) \cdot 0}{0}\right)\right)\right) \]

      16. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(y \cdot \frac{1}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      17. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + y \cdot \frac{1}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      18. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot 0}{0}\right)\right)\right) \]

      19. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\left(\frac{y}{2} + \frac{y}{2}\right) \cdot \left(\frac{y}{2} - \frac{y}{2}\right)}{0}\right)\right)\right) \]

      20. difference-of-squares N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{0}\right)\right)\right) \]

      21. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{\frac{y}{2} \cdot \frac{y}{2} - \frac{y}{2} \cdot \frac{y}{2}}{\frac{y}{2} - \frac{y}{2}}\right)\right)\right) \]

      22. flip-+ N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(\frac{y}{2} + \frac{y}{2}\right)\right)\right) \]

      23. count-2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{exp.f64}\left(\left(2 \cdot \frac{y}{2}\right)\right)\right) \]

   4. Applied egg-rr 52.0%

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

Alternative 3: 93.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x + x \cdot \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.0005)
   (+ x (* x (* (* y y) (+ 1.0 (* y (* y 0.5))))))
   (* x (* (* y y) (* y (* y (* y (* y 0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x + (x * ((y * y) * (1.0 + (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	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) <= 0.0005d0) then
        tmp = x + (x * ((y * y) * (1.0d0 + (y * (y * 0.5d0)))))
    else
        tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x + (x * ((y * y) * (1.0 + (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x + (x * ((y * y) * (1.0 + (y * (y * 0.5)))))
	else:
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0005)
		tmp = Float64(x + Float64(x * Float64(Float64(y * y) * Float64(1.0 + Float64(y * Float64(y * 0.5))))));
	else
		tmp = Float64(x * Float64(Float64(y * y) * Float64(y * Float64(y * Float64(y * Float64(y * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.0005)
		tmp = x + (x * ((y * y) * (1.0 + (y * (y * 0.5)))));
	else
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], N[(x + N[(x * N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right)\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(1 + y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(1 + y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      10. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   7. Applied egg-rr 99.8%

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

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 93.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 93.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{6}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       3. cube-prod N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left(y \cdot y\right)}^{\color{blue}{3}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left({y}^{2}\right)}^{3}\right)\right) \]

       5. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{y}}^{2}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{4} \cdot {y}^{2}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{4}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       14. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right)\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       22. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

       26. *-lowering-*.f64 93.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 93.4%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x + x \cdot \left(\left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \left(y \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.0005)
   (* x (+ 1.0 (* (* y y) (+ 1.0 (* y (* y 0.5))))))
   (* x (* (* y y) (* y (* y (* y (* y 0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	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) <= 0.0005d0) then
        tmp = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * 0.5d0)))))
    else
        tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * 0.5)))));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * 0.5)))))
	else:
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0005)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(y * y) * Float64(1.0 + Float64(y * Float64(y * 0.5))))));
	else
		tmp = Float64(x * Float64(Float64(y * y) * Float64(y * Float64(y * Float64(y * Float64(y * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.0005)
		tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * 0.5)))));
	else
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], N[(x * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(1.0 + N[(y * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

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

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 93.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 93.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{6}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       3. cube-prod N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left(y \cdot y\right)}^{\color{blue}{3}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left({y}^{2}\right)}^{3}\right)\right) \]

       5. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{y}}^{2}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{4} \cdot {y}^{2}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{4}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       14. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right)\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       22. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

       26. *-lowering-*.f64 93.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 93.4%

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

Alternative 5: 93.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;\frac{x}{\frac{1}{y \cdot y + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.0005)
   (/ x (/ 1.0 (+ (* y y) 1.0)))
   (* x (* (* y y) (* y (* y (* y (* y 0.16666666666666666))))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x / (1.0 / ((y * y) + 1.0));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	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) <= 0.0005d0) then
        tmp = x / (1.0d0 / ((y * y) + 1.0d0))
    else
        tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x / (1.0 / ((y * y) + 1.0));
	} else {
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x / (1.0 / ((y * y) + 1.0))
	else:
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0005)
		tmp = Float64(x / Float64(1.0 / Float64(Float64(y * y) + 1.0)));
	else
		tmp = Float64(x * Float64(Float64(y * y) * Float64(y * Float64(y * Float64(y * Float64(y * 0.16666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.0005)
		tmp = x / (1.0 / ((y * y) + 1.0));
	else
		tmp = x * ((y * y) * (y * (y * (y * (y * 0.16666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], N[(x / N[(1.0 / N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e-4

   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. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{x} \cdot {y}^{2} \]

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 99.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1}{\color{blue}{y \cdot y - 1}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1\right) \cdot \color{blue}{\frac{1}{y \cdot y - 1}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \frac{1}{y \cdot y - 1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1}{\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1} \cdot \frac{\color{blue}{1}}{y \cdot y - 1}\right)\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1}{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1\right), \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)\right)}\right)\right) \]

   7. Applied egg-rr 99.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}}}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\frac{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1}}{\color{blue}{y \cdot y + -1}}}\right)\right) \]

   9. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + y \cdot y}}} \]

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 93.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)}\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right)\right)\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 93.4%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. Simplified 93.4%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} \cdot {y}^{6}\right)}\right) \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{3}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{3} \cdot \color{blue}{{y}^{3}}\right)\right)\right) \]

       3. cube-prod N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left(y \cdot y\right)}^{\color{blue}{3}}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot {\left({y}^{2}\right)}^{3}\right)\right) \]

       5. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(\left({y}^{2} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       6. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{\left(2 \cdot 2\right)} \cdot {\color{blue}{y}}^{2}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left({y}^{4} \cdot {y}^{2}\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{4}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} \cdot {y}^{4}\right)}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} \cdot {y}^{4}\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

       14. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{6} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

       15. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot {y}^{2}\right)\right)\right)\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

       21. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right)\right) \]

       22. associate-*r* N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

       23. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right) \]

       25. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

       26. *-lowering-*.f64 93.4%

           \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 93.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;\frac{x}{\frac{1}{y \cdot y + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * (0.5d0 + ((y * y) * 0.16666666666666666d0)))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (0.5 + ((y * y) * 0.16666666666666666)))))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Simplified 96.7%

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

Alternative 7: 93.8% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + ((y * y) * (1.0d0 + (y * (y * (y * (y * 0.16666666666666666d0)))))))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (y * (y * 0.16666666666666666)))))));
}

Python

def code(x, y):
	return x * (1.0 + ((y * y) * (1.0 + (y * (y * (y * (y * 0.16666666666666666)))))))

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + ((y * y) * (1.0 + (y * (y * (y * (y * 0.16666666666666666)))))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

5. Simplified 96.7%

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

6. Taylor expanded in y around inf

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot {y}^{3}\right)}\right)\right)\right)\right)\right) \]
7. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{1}{6} \cdot \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(\frac{1}{6} \cdot y\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} \cdot y\right)\right)\right)\right)\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} \cdot y\right)\right)}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right)\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 96.3%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 96.3%

   \[\leadsto x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(1 + y \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\right)}\right)\right) \]
9. Add Preprocessing

Alternative 8: 91.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;\frac{x}{\frac{1}{y \cdot y + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 0.0005)
   (/ x (/ 1.0 (+ (* y y) 1.0)))
   (* x (* y (* 0.5 (* y (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x / (1.0 / ((y * y) + 1.0));
	} else {
		tmp = x * (y * (0.5 * (y * (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) <= 0.0005d0) then
        tmp = x / (1.0d0 / ((y * y) + 1.0d0))
    else
        tmp = x * (y * (0.5d0 * (y * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		tmp = x / (1.0 / ((y * y) + 1.0));
	} else {
		tmp = x * (y * (0.5 * (y * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x / (1.0 / ((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) <= 0.0005)
		tmp = Float64(x / Float64(1.0 / Float64(Float64(y * y) + 1.0)));
	else
		tmp = Float64(x * Float64(y * Float64(0.5 * Float64(y * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 0.0005)
		tmp = x / (1.0 / ((y * y) + 1.0));
	else
		tmp = x * (y * (0.5 * (y * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], N[(x / N[(1.0 / N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $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) < 5.0000000000000001e-4

   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. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{x} \cdot {y}^{2} \]

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 99.2%

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

      1. flip-+ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1}{\color{blue}{y \cdot y - 1}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1\right) \cdot \color{blue}{\frac{1}{y \cdot y - 1}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \frac{1}{y \cdot y - 1}\right)\right) \]

      4. flip-- N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1}{\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1} \cdot \frac{\color{blue}{1}}{y \cdot y - 1}\right)\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1}{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1\right), \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)\right)}\right)\right) \]

   7. Applied egg-rr 99.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}}}\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\frac{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1}}{\color{blue}{y \cdot y + -1}}}\right)\right) \]

   9. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + y \cdot y}}} \]

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. distribute-lft-out N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. distribute-lft-in N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(1 + \frac{1}{2} \cdot {y}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(1 + \frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{1} + \frac{1}{2} \cdot {y}^{2}\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

      18. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 88.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 88.8%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{2} \cdot {y}^{2}\right)\right)}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot y\right) \cdot \color{blue}{y}\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\left(\frac{1}{2} \cdot {y}^{2}\right) \cdot y\right)}\right)\right) \]

      13. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \color{blue}{\left({y}^{2} \cdot y\right)}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot \left(\left(y \cdot y\right) \cdot y\right)\right)\right)\right) \]

      15. unpow3 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{1}{2} \cdot {y}^{\color{blue}{3}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({y}^{3}\right)}\right)\right)\right) \]

      17. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot \color{blue}{\left(y \cdot y\right)}\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \left(y \cdot {y}^{\color{blue}{2}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2}\right)}\right)\right)\right)\right) \]

      20. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{y}\right)\right)\right)\right)\right) \]

      21. *-lowering-*.f64 88.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right)\right) \]

   8. Simplified 88.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;\frac{x}{\frac{1}{y \cdot y + 1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \left(0.5 \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.0005) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		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) <= 0.0005d0) 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) <= 0.0005) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0005)
		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) <= 0.0005)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e-4

   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 97.9%

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

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-rgt-identity N/A

         \[\leadsto x \cdot 1 + \color{blue}{x} \cdot {y}^{2} \]

      2. distribute-lft-in N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

      7. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

   5. Simplified 63.5%

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

   6. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({y}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(y \cdot \color{blue}{y}\right)\right) \]

      3. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   8. Simplified 63.5%

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

Alternative 10: 82.5% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0

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

   1. *-rgt-identity N/A

      \[\leadsto x \cdot 1 + \color{blue}{x} \cdot {y}^{2} \]

   2. distribute-lft-in N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

   7. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

5. Simplified 81.3%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1}{\color{blue}{y \cdot y - 1}}\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1 \cdot 1\right) \cdot \color{blue}{\frac{1}{y \cdot y - 1}}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) - 1\right) \cdot \frac{1}{y \cdot y - 1}\right)\right) \]

   4. flip-- N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1}{\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1} \cdot \frac{\color{blue}{1}}{y \cdot y - 1}\right)\right) \]

   5. frac-times N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1}{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right) - 1 \cdot 1\right) \cdot 1\right), \color{blue}{\left(\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1\right) \cdot \left(y \cdot y - 1\right)\right)}\right)\right) \]

7. Applied egg-rr 51.3%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}\right)}\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1\right) \cdot \left(y \cdot y + -1\right)}}}\right)\right) \]

   5. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\frac{\frac{\left(\left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) \cdot \left(y \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - 1\right) \cdot 1}{y \cdot \left(y \cdot \left(y \cdot y\right)\right) + 1}}{\color{blue}{y \cdot y + -1}}}\right)\right) \]

9. Applied egg-rr 81.3%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + y \cdot y}}} \]

10. Final simplification 81.3%

    \[\leadsto \frac{x}{\frac{1}{y \cdot y + 1}} \]
11. Add Preprocessing

Alternative 11: 82.5% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-rgt-identity N/A

      \[\leadsto x \cdot 1 + \color{blue}{x} \cdot {y}^{2} \]

   2. distribute-lft-in N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {y}^{2}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({y}^{2} + \color{blue}{1}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left({y}^{2}\right), \color{blue}{1}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(y \cdot y\right), 1\right)\right) \]

   7. *-lowering-*.f64 81.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, y\right), 1\right)\right) \]

5. Simplified 81.3%

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

Alternative 12: 51.0% accurate, 105.0× 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 51.0%

   \[\leadsto \color{blue}{x} \]
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 2024138 
(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))))