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

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 6

Speedup: 1.0×

• 49.4% 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, 0.5× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (pow (exp (+ y y)) (* y 0.5))))

C

double code(double x, double y) {
	return x * pow(exp((y + y)), (y * 0.5));
}

Fortran

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

Java

public static double code(double x, double y) {
	return x * Math.pow(Math.exp((y + y)), (y * 0.5));
}

Python

def code(x, y):
	return x * math.pow(math.exp((y + y)), (y * 0.5))

Julia

function code(x, y)
	return Float64(x * (exp(Float64(y + y)) ^ Float64(y * 0.5)))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (exp((y + y)) ^ (y * 0.5));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. pow-exp 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. div-inv 100.0%

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

   6. metadata-eval 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

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 100.0%

   \[x \cdot e^{y \cdot y} \]
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: 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. Final simplification 100.0%

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

Alternative 4: 81.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 2e-15) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-15) {
		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) <= 2d-15) 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) <= 2e-15) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-15:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-15)
		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) <= 2e-15)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-15], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.0000000000000002e-15

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

   if 2.0000000000000002e-15 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.9%

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

      1. unpow2 65.9%

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

   4. Simplified 65.9%

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

   5. Taylor expanded in y around inf 65.9%

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

      1. unpow2 65.9%

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

      2. *-commutative 65.9%

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

   7. Simplified 65.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5: 82.1% 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. Taylor expanded in y around 0 83.2%

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

   1. unpow2 83.2%

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

4. Simplified 83.2%

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

5. Final simplification 83.2%

   \[\leadsto x \cdot \left(y \cdot y + 1\right) \]

Alternative 6: 51.6% 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. Taylor expanded in y around 0 52.6%

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

3. Final simplification 52.6%

   \[\leadsto x \]

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 2023224 
(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))))