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

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

Alternatives: 7

Speedup: 1.0×

• 50.5% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)}\\ x \cdot \left(t_0 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (pow (pow (exp y) 0.25) (* y 2.0)))) (* x (* t_0 t_0))))

C

double code(double x, double y) {
	double t_0 = pow(pow(exp(y), 0.25), (y * 2.0));
	return x * (t_0 * t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (exp(y) ** 0.25d0) ** (y * 2.0d0)
    code = x * (t_0 * t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(Math.pow(Math.exp(y), 0.25), (y * 2.0));
	return x * (t_0 * t_0);
}

Python

def code(x, y):
	t_0 = math.pow(math.pow(math.exp(y), 0.25), (y * 2.0))
	return x * (t_0 * t_0)

Julia

function code(x, y)
	t_0 = (exp(y) ^ 0.25) ^ Float64(y * 2.0)
	return Float64(x * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (exp(y) ^ 0.25) ^ (y * 2.0);
	tmp = x * (t_0 * t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Power[N[Power[N[Exp[y], $MachinePrecision], 0.25], $MachinePrecision], N[(y * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

   1. exp-prod 100.0%

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

   2. sqr-pow 100.0%

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

   3. pow-prod-down 100.0%

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

   4. prod-exp 100.0%

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

   5. count-2 100.0%

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

   6. *-commutative 100.0%

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

   7. div-inv 100.0%

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

   8. metadata-eval 100.0%

      \[\leadsto x \cdot {\left(e^{y \cdot 2}\right)}^{\left(y \cdot \color{blue}{0.5}\right)} \]

3. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{{\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.5\right)}} \]
4. Step-by-step derivation

   1. pow-exp 99.9%

      \[\leadsto x \cdot \color{blue}{e^{\left(y \cdot 2\right) \cdot \left(y \cdot 0.5\right)}} \]

   2. *-commutative 99.9%

      \[\leadsto x \cdot e^{\color{blue}{\left(y \cdot 0.5\right) \cdot \left(y \cdot 2\right)}} \]

   3. add-log-exp 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\log \left(e^{y \cdot 0.5}\right)} \cdot \left(y \cdot 2\right)} \]

   4. exp-prod 100.0%

      \[\leadsto x \cdot e^{\log \color{blue}{\left({\left(e^{y}\right)}^{0.5}\right)} \cdot \left(y \cdot 2\right)} \]

   5. pow1/2 100.0%

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

   6. pow-to-exp 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. unpow-prod-down 100.0%

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

   9. pow1/2 100.0%

      \[\leadsto x \cdot \left({\left(\sqrt{\color{blue}{{\left(e^{y}\right)}^{0.5}}}\right)}^{\left(y \cdot 2\right)} \cdot {\left(\sqrt{\sqrt{e^{y}}}\right)}^{\left(y \cdot 2\right)}\right) \]

   10. sqrt-pow1 100.0%

       \[\leadsto x \cdot \left({\color{blue}{\left({\left(e^{y}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{\left(y \cdot 2\right)} \cdot {\left(\sqrt{\sqrt{e^{y}}}\right)}^{\left(y \cdot 2\right)}\right) \]

   11. metadata-eval 100.0%

       \[\leadsto x \cdot \left({\left({\left(e^{y}\right)}^{\color{blue}{0.25}}\right)}^{\left(y \cdot 2\right)} \cdot {\left(\sqrt{\sqrt{e^{y}}}\right)}^{\left(y \cdot 2\right)}\right) \]

   12. pow1/2 100.0%

       \[\leadsto x \cdot \left({\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)} \cdot {\left(\sqrt{\color{blue}{{\left(e^{y}\right)}^{0.5}}}\right)}^{\left(y \cdot 2\right)}\right) \]

   13. sqrt-pow1 100.0%

       \[\leadsto x \cdot \left({\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)} \cdot {\color{blue}{\left({\left(e^{y}\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{\left(y \cdot 2\right)}\right) \]

   14. metadata-eval 100.0%

       \[\leadsto x \cdot \left({\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)} \cdot {\left({\left(e^{y}\right)}^{\color{blue}{0.25}}\right)}^{\left(y \cdot 2\right)}\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left({\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)} \cdot {\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)}\right)} \]

6. Final simplification 100.0%

   \[\leadsto x \cdot \left({\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)} \cdot {\left({\left(e^{y}\right)}^{0.25}\right)}^{\left(y \cdot 2\right)}\right) \]

Alternative 2: 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]

Derivation

1. Initial program 99.9%

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

   1. exp-prod 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+159} \lor \neg \left(y \leq 10^{+142}\right):\\ \;\;\;\;x \cdot {y}^{2}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.3e+159) (not (<= y 1e+142)))
   (* x (pow y 2.0))
   (+ x (* y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+159) || !(y <= 1e+142)) {
		tmp = x * pow(y, 2.0);
	} else {
		tmp = x + (y * (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.3d+159)) .or. (.not. (y <= 1d+142))) then
        tmp = x * (y ** 2.0d0)
    else
        tmp = x + (y * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.3e+159) || !(y <= 1e+142)) {
		tmp = x * Math.pow(y, 2.0);
	} else {
		tmp = x + (y * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.3e+159) or not (y <= 1e+142):
		tmp = x * math.pow(y, 2.0)
	else:
		tmp = x + (y * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.3e+159) || !(y <= 1e+142))
		tmp = Float64(x * (y ^ 2.0));
	else
		tmp = Float64(x + Float64(y * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.3e+159) || ~((y <= 1e+142)))
		tmp = x * (y ^ 2.0);
	else
		tmp = x + (y * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.3e+159], N[Not[LessEqual[y, 1e+142]], $MachinePrecision]], N[(x * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999999e159 or 1.00000000000000005e142 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.8%

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

      1. *-commutative 98.8%

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

   4. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. unpow2 98.8%

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

      3. associate-*r* 76.9%

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

      4. add-log-exp 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. log-prod 100.0%

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

      7. metadata-eval 100.0%

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

      8. add-log-exp 76.9%

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

      9. distribute-rgt-in 38.0%

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

   6. Applied egg-rr 38.0%

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

   7. Taylor expanded in x around 0 76.9%

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

   8. Taylor expanded in y around inf 98.8%

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

   if -3.2999999999999999e159 < y < 1.00000000000000005e142

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 76.8%

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

      1. *-commutative 76.8%

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

   4. Simplified 76.8%

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

      1. *-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. associate-*r* 76.8%

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

      4. add-log-exp 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. log-prod 98.6%

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

      7. metadata-eval 98.6%

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

      8. add-log-exp 76.8%

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

      9. distribute-rgt-in 72.3%

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

   6. Applied egg-rr 72.3%

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

   7. Taylor expanded in x around 0 76.8%

      \[\leadsto x + \left(\color{blue}{0} + y \cdot \left(x \cdot y\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+159} \lor \neg \left(y \leq 10^{+142}\right):\\ \;\;\;\;x \cdot {y}^{2}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 75.1% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 83.4%

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

   1. *-commutative 83.4%

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

4. Simplified 83.4%

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

   1. *-commutative 83.4%

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

   2. unpow2 83.4%

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

   3. associate-*r* 76.8%

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

   4. add-log-exp 99.0%

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

   5. *-un-lft-identity 99.0%

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

   6. log-prod 99.0%

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

   7. metadata-eval 99.0%

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

   8. add-log-exp 76.8%

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

   9. distribute-rgt-in 62.0%

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

6. Applied egg-rr 62.0%

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

7. Taylor expanded in x around 0 76.8%

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

8. Final simplification 76.8%

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

Alternative 6: 55.8% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 83.4%

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

   1. *-commutative 83.4%

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

4. Simplified 83.4%

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

   1. *-commutative 83.4%

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

   2. unpow2 83.4%

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

   3. associate-*r* 76.8%

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

   4. *-un-lft-identity 76.8%

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

   5. *-commutative 76.8%

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

   6. metadata-eval 76.8%

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

   7. distribute-lft-out 76.8%

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

   8. distribute-rgt-in 76.8%

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

6. Applied egg-rr 76.8%

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

   1. flip-+ 0.0%

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

   2. +-inverses 0.0%

      \[\leadsto x + \frac{\color{blue}{0}}{\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right) - \left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)} \]

   3. metadata-eval 0.0%

      \[\leadsto x + \frac{\color{blue}{{0}^{0.5}}}{\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right) - \left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)} \]

   4. mul0-lft 0.0%

      \[\leadsto x + \frac{{\color{blue}{\left(0 \cdot \left(x \cdot y\right)\right)}}^{0.5}}{\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right) - \left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)} \]

   5. pow1/2 0.0%

      \[\leadsto x + \frac{\color{blue}{\sqrt{0 \cdot \left(x \cdot y\right)}}}{\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right) - \left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)} \]

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

      \[\leadsto x + \frac{\sqrt{0 \cdot \left(x \cdot y\right)}}{\color{blue}{{0}^{0.5}}} \]

   8. mul0-lft 0.0%

      \[\leadsto x + \frac{\sqrt{0 \cdot \left(x \cdot y\right)}}{{\color{blue}{\left(0 \cdot \left(x \cdot y\right)\right)}}^{0.5}} \]

   9. pow1/2 0.0%

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

   10. sqrt-div 0.0%

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

   11. mul0-lft 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto x + \sqrt{\frac{\color{blue}{\left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) \cdot \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) - \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) \cdot \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right)}}{0 \cdot \left(x \cdot y\right)}} \]

   13. mul0-lft 0.0%

       \[\leadsto x + \sqrt{\frac{\left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) \cdot \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) - \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right) \cdot \left(\left(y \cdot 0.5\right) \cdot \left(x \cdot y\right)\right)}{\color{blue}{0}}} \]

   14. +-inverses 0.0%

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

   15. flip-+ 47.2%

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

   16. *-un-lft-identity 47.2%

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

8. Applied egg-rr 53.9%

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

9. Final simplification 53.9%

   \[\leadsto x + x \cdot y \]

Alternative 7: 50.9% 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 99.9%

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

2. Taylor expanded in y around 0 48.1%

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

3. Final simplification 48.1%

   \[\leadsto x \]

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 2023301 
(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))))