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

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 6

Speedup: 1.0×

• 50.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

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 99.6%

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

2. Final simplification 99.6%

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

Alternative 2: 87.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y \cdot y} + -1\\ t_1 := 1 - y \cdot y\\ t_2 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x + t_2\\ \mathbf{elif}\;y \cdot y \leq 10^{+296}:\\ \;\;\;\;\frac{x}{t_0} \cdot \frac{t_0}{t_1} + \left(t_2 \cdot \left(y \cdot y + -1\right)\right) \cdot \frac{\frac{y \cdot y}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (* y y)) -1.0))
        (t_1 (- 1.0 (* y y)))
        (t_2 (* x (* y y))))
   (if (<= (* y y) 2e+42)
     (+ x t_2)
     (if (<= (* y y) 1e+296)
       (+
        (* (/ x t_0) (/ t_0 t_1))
        (* (* t_2 (+ (* y y) -1.0)) (/ (/ (* y y) t_1) t_1)))
       t_2))))

C

double code(double x, double y) {
	double t_0 = (1.0 / (y * y)) + -1.0;
	double t_1 = 1.0 - (y * y);
	double t_2 = x * (y * y);
	double tmp;
	if ((y * y) <= 2e+42) {
		tmp = x + t_2;
	} else if ((y * y) <= 1e+296) {
		tmp = ((x / t_0) * (t_0 / t_1)) + ((t_2 * ((y * y) + -1.0)) * (((y * y) / t_1) / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 / (y * y)) + (-1.0d0)
    t_1 = 1.0d0 - (y * y)
    t_2 = x * (y * y)
    if ((y * y) <= 2d+42) then
        tmp = x + t_2
    else if ((y * y) <= 1d+296) then
        tmp = ((x / t_0) * (t_0 / t_1)) + ((t_2 * ((y * y) + (-1.0d0))) * (((y * y) / t_1) / t_1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / (y * y)) + -1.0;
	double t_1 = 1.0 - (y * y);
	double t_2 = x * (y * y);
	double tmp;
	if ((y * y) <= 2e+42) {
		tmp = x + t_2;
	} else if ((y * y) <= 1e+296) {
		tmp = ((x / t_0) * (t_0 / t_1)) + ((t_2 * ((y * y) + -1.0)) * (((y * y) / t_1) / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / (y * y)) + -1.0
	t_1 = 1.0 - (y * y)
	t_2 = x * (y * y)
	tmp = 0
	if (y * y) <= 2e+42:
		tmp = x + t_2
	elif (y * y) <= 1e+296:
		tmp = ((x / t_0) * (t_0 / t_1)) + ((t_2 * ((y * y) + -1.0)) * (((y * y) / t_1) / t_1))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / Float64(y * y)) + -1.0)
	t_1 = Float64(1.0 - Float64(y * y))
	t_2 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * y) <= 2e+42)
		tmp = Float64(x + t_2);
	elseif (Float64(y * y) <= 1e+296)
		tmp = Float64(Float64(Float64(x / t_0) * Float64(t_0 / t_1)) + Float64(Float64(t_2 * Float64(Float64(y * y) + -1.0)) * Float64(Float64(Float64(y * y) / t_1) / t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / (y * y)) + -1.0;
	t_1 = 1.0 - (y * y);
	t_2 = x * (y * y);
	tmp = 0.0;
	if ((y * y) <= 2e+42)
		tmp = x + t_2;
	elseif ((y * y) <= 1e+296)
		tmp = ((x / t_0) * (t_0 / t_1)) + ((t_2 * ((y * y) + -1.0)) * (((y * y) / t_1) / t_1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 2e+42], N[(x + t$95$2), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1e+296], N[(N[(N[(x / t$95$0), $MachinePrecision] * N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[(N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2.00000000000000009e42

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 94.0%

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

      1. unpow2 94.0%

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

   4. Simplified 94.0%

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

   if 2.00000000000000009e42 < (*.f64 y y) < 9.99999999999999981e295

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 26.6%

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

      1. unpow2 26.6%

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

   4. Simplified 26.6%

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

      1. flip-+ 24.9%

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

      2. div-sub 24.9%

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

      3. associate-/l* 29.2%

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

      4. associate-/l* 3.2%

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

      5. frac-sub 33.3%

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

   6. Applied egg-rr 33.3%

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

      1. div-sub 33.3%

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

      2. div-inv 33.3%

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

      3. cancel-sign-sub-inv 33.3%

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

   8. Applied egg-rr 56.7%

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

   if 9.99999999999999981e295 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

      1. flip-+ 0.0%

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

      2. div-sub 0.0%

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

      3. associate-/l* 0.0%

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

      4. associate-/l* 0.0%

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

      5. frac-sub 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \cdot y \leq 10^{+296}:\\ \;\;\;\;\frac{x}{\frac{1}{y \cdot y} + -1} \cdot \frac{\frac{1}{y \cdot y} + -1}{1 - y \cdot y} + \left(\left(x \cdot \left(y \cdot y\right)\right) \cdot \left(y \cdot y + -1\right)\right) \cdot \frac{\frac{y \cdot y}{1 - y \cdot y}}{1 - y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 3: 85.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y \cdot y}\\ t_1 := x \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x + t_1\\ \mathbf{elif}\;y \cdot y \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\frac{1}{\frac{x - y \cdot \left(x \cdot y\right)}{\frac{x}{\frac{t_0 + -1}{x \cdot \left(t_0 + \left(-1 + \left(y \cdot y\right) \cdot \left(y \cdot y + -1\right)\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;x + t_1 \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* y y))) (t_1 (* x (* y y))))
   (if (<= (* y y) 2e+42)
     (+ x t_1)
     (if (<= (* y y) 5e+249)
       (/
        1.0
        (/
         (- x (* y (* x y)))
         (/
          x
          (/
           (+ t_0 -1.0)
           (* x (+ t_0 (+ -1.0 (* (* y y) (+ (* y y) -1.0)))))))))
       (+ x (* t_1 2.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (y * y);
	double t_1 = x * (y * y);
	double tmp;
	if ((y * y) <= 2e+42) {
		tmp = x + t_1;
	} else if ((y * y) <= 5e+249) {
		tmp = 1.0 / ((x - (y * (x * y))) / (x / ((t_0 + -1.0) / (x * (t_0 + (-1.0 + ((y * y) * ((y * y) + -1.0))))))));
	} else {
		tmp = x + (t_1 * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (y * y)
    t_1 = x * (y * y)
    if ((y * y) <= 2d+42) then
        tmp = x + t_1
    else if ((y * y) <= 5d+249) then
        tmp = 1.0d0 / ((x - (y * (x * y))) / (x / ((t_0 + (-1.0d0)) / (x * (t_0 + ((-1.0d0) + ((y * y) * ((y * y) + (-1.0d0)))))))))
    else
        tmp = x + (t_1 * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (y * y);
	double t_1 = x * (y * y);
	double tmp;
	if ((y * y) <= 2e+42) {
		tmp = x + t_1;
	} else if ((y * y) <= 5e+249) {
		tmp = 1.0 / ((x - (y * (x * y))) / (x / ((t_0 + -1.0) / (x * (t_0 + (-1.0 + ((y * y) * ((y * y) + -1.0))))))));
	} else {
		tmp = x + (t_1 * 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (y * y)
	t_1 = x * (y * y)
	tmp = 0
	if (y * y) <= 2e+42:
		tmp = x + t_1
	elif (y * y) <= 5e+249:
		tmp = 1.0 / ((x - (y * (x * y))) / (x / ((t_0 + -1.0) / (x * (t_0 + (-1.0 + ((y * y) * ((y * y) + -1.0))))))))
	else:
		tmp = x + (t_1 * 2.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(y * y))
	t_1 = Float64(x * Float64(y * y))
	tmp = 0.0
	if (Float64(y * y) <= 2e+42)
		tmp = Float64(x + t_1);
	elseif (Float64(y * y) <= 5e+249)
		tmp = Float64(1.0 / Float64(Float64(x - Float64(y * Float64(x * y))) / Float64(x / Float64(Float64(t_0 + -1.0) / Float64(x * Float64(t_0 + Float64(-1.0 + Float64(Float64(y * y) * Float64(Float64(y * y) + -1.0)))))))));
	else
		tmp = Float64(x + Float64(t_1 * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (y * y);
	t_1 = x * (y * y);
	tmp = 0.0;
	if ((y * y) <= 2e+42)
		tmp = x + t_1;
	elseif ((y * y) <= 5e+249)
		tmp = 1.0 / ((x - (y * (x * y))) / (x / ((t_0 + -1.0) / (x * (t_0 + (-1.0 + ((y * y) * ((y * y) + -1.0))))))));
	else
		tmp = x + (t_1 * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 2e+42], N[(x + t$95$1), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 5e+249], N[(1.0 / N[(N[(x - N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(x * N[(t$95$0 + N[(-1.0 + N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$1 * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2.00000000000000009e42

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 94.0%

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

      1. unpow2 94.0%

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

   4. Simplified 94.0%

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

   if 2.00000000000000009e42 < (*.f64 y y) < 4.9999999999999996e249

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 18.5%

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

      1. unpow2 18.5%

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

   4. Simplified 18.5%

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

      1. flip-+ 26.7%

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

      2. div-sub 26.7%

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

      3. associate-/l* 3.1%

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

      4. frac-sub 26.7%

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

   6. Applied egg-rr 26.7%

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

      1. clear-num 26.7%

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

      2. inv-pow 26.7%

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

      3. metadata-eval 26.7%

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

      4. sqr-pow 10.8%

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

   8. Applied egg-rr 18.1%

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

      1. pow-sqr 36.4%

         \[\leadsto \color{blue}{{\left(\frac{x - x \cdot \left(y \cdot y\right)}{\frac{x \cdot \left(x \cdot \left(\frac{1}{y \cdot y} + -1\right) - \left(y \cdot y\right) \cdot \left(x - x \cdot \left(y \cdot y\right)\right)\right)}{\frac{1}{y \cdot y} + -1}}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 36.4%

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

      3. unpow-1 36.4%

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

      4. associate-*r* 36.4%

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

      5. associate-/l* 36.4%

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

   10. Simplified 36.4%

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

   11. Taylor expanded in x around 0 51.0%

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

       1. sub-neg 51.0%

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

       2. unpow2 51.0%

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

       3. metadata-eval 51.0%

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

       4. +-commutative 51.0%

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

       5. unpow2 51.0%

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

       6. unpow2 51.0%

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

       7. unpow2 51.0%

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

   13. Simplified 51.0%

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

   if 4.9999999999999996e249 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 97.6%

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

      1. unpow2 97.6%

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

   4. Simplified 97.6%

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

      1. flip-+ 1.3%

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

      2. div-sub 1.3%

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

      3. associate-/l* 1.3%

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

      4. associate-/l* 0.1%

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

      5. frac-sub 2.5%

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

   6. Applied egg-rr 2.5%

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

   7. Taylor expanded in y around 0 2.5%

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

      1. unpow2 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in y around 0 97.6%

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

       1. unpow2 97.6%

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

   12. Simplified 97.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;y \cdot y \leq 5 \cdot 10^{+249}:\\ \;\;\;\;\frac{1}{\frac{x - y \cdot \left(x \cdot y\right)}{\frac{x}{\frac{\frac{1}{y \cdot y} + -1}{x \cdot \left(\frac{1}{y \cdot y} + \left(-1 + \left(y \cdot y\right) \cdot \left(y \cdot y + -1\right)\right)\right)}}}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(x \cdot \left(y \cdot y\right)\right) \cdot 2\\ \end{array} \]

Alternative 4: 81.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.0002) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0002) {
		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) <= 0.0002d0) 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) <= 0.0002) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0002:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0002)
		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) <= 0.0002)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0002], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.0000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

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

   if 2.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 99.3%

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

   2. Taylor expanded in y around 0 67.0%

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

      1. unpow2 67.0%

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

   4. Simplified 67.0%

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

      1. flip-+ 9.3%

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

      2. div-sub 9.3%

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

      3. associate-/l* 10.9%

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

      4. associate-/l* 1.3%

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

      5. frac-sub 12.4%

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

   6. Applied egg-rr 12.4%

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

   7. Taylor expanded in y around inf 67.0%

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

      1. unpow2 67.0%

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

   9. Simplified 67.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5: 81.4% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in y around 0 83.3%

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

   1. unpow2 83.3%

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

4. Simplified 83.3%

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

5. Final simplification 83.3%

   \[\leadsto x + x \cdot \left(y \cdot y\right) \]

Alternative 6: 50.4% 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 99.6%

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

2. Taylor expanded in y around 0 51.5%

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

3. Final simplification 51.5%

   \[\leadsto x \]

Developer target: 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 2023297 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :herbie-target
  (* x (pow (exp y) y))

  (* x (exp (* y y))))