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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 4

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: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0002 \lor \neg \left(y \cdot y \leq 10^{+296}\right):\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 0.0002) (not (<= (* y y) 1e+296)))
   (+ x (* x (* y y)))
   (exp (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 0.0002) || !((y * y) <= 1e+296)) {
		tmp = x + (x * (y * y));
	} else {
		tmp = exp((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) .or. (.not. ((y * y) <= 1d+296))) then
        tmp = x + (x * (y * y))
    else
        tmp = exp((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y * y) <= 0.0002) || !((y * y) <= 1e+296)) {
		tmp = x + (x * (y * y));
	} else {
		tmp = Math.exp((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 0.0002) or not ((y * y) <= 1e+296):
		tmp = x + (x * (y * y))
	else:
		tmp = math.exp((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 0.0002) || !(Float64(y * y) <= 1e+296))
		tmp = Float64(x + Float64(x * Float64(y * y)));
	else
		tmp = exp(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y * y) <= 0.0002) || ~(((y * y) <= 1e+296)))
		tmp = x + (x * (y * y));
	else
		tmp = exp((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 0.0002], N[Not[LessEqual[N[(y * y), $MachinePrecision], 1e+296]], $MachinePrecision]], N[(x + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.0000000000000001e-4 or 9.99999999999999981e295 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.8%

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

      1. unpow2 99.8%

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

   4. Simplified 99.8%

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

   if 2.0000000000000001e-4 < (*.f64 y y) < 9.99999999999999981e295

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

      2. add-exp-log 51.1%

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

      3. prod-exp 52.5%

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

   3. Applied egg-rr 52.5%

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

   4. Taylor expanded in y around inf 51.1%

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

      1. unpow2 51.1%

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

   6. Simplified 51.1%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0002 \lor \neg \left(y \cdot y \leq 10^{+296}\right):\\ \;\;\;\;x + x \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y \cdot y}\\ \end{array} \]

Alternative 3: 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 4: 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))))