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

Percentage Accurate: 100.0% → 100.0%

Time: 3.7s

Alternatives: 6

Speedup: 1.0×

• 49.6% 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

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

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 2: 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 3: 81.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+120} \lor \neg \left(y \cdot y \leq 10^{+194}\right):\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - t_0 \cdot t_0}{x - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (or (<= (* y y) 5e+120) (not (<= (* y y) 1e+194)))
     (* x (+ (* y y) 1.0))
     (/ (- (* x x) (* t_0 t_0)) (- x t_0)))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (((y * y) <= 5e+120) || !((y * y) <= 1e+194)) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = ((x * x) - (t_0 * t_0)) / (x - t_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) :: tmp
    t_0 = y * (x * y)
    if (((y * y) <= 5d+120) .or. (.not. ((y * y) <= 1d+194))) then
        tmp = x * ((y * y) + 1.0d0)
    else
        tmp = ((x * x) - (t_0 * t_0)) / (x - t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if (((y * y) <= 5e+120) || !((y * y) <= 1e+194)) {
		tmp = x * ((y * y) + 1.0);
	} else {
		tmp = ((x * x) - (t_0 * t_0)) / (x - t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if ((y * y) <= 5e+120) or not ((y * y) <= 1e+194):
		tmp = x * ((y * y) + 1.0)
	else:
		tmp = ((x * x) - (t_0 * t_0)) / (x - t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if ((Float64(y * y) <= 5e+120) || !(Float64(y * y) <= 1e+194))
		tmp = Float64(x * Float64(Float64(y * y) + 1.0));
	else
		tmp = Float64(Float64(Float64(x * x) - Float64(t_0 * t_0)) / Float64(x - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if (((y * y) <= 5e+120) || ~(((y * y) <= 1e+194)))
		tmp = x * ((y * y) + 1.0);
	else
		tmp = ((x * x) - (t_0 * t_0)) / (x - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(y * y), $MachinePrecision], 5e+120], N[Not[LessEqual[N[(y * y), $MachinePrecision], 1e+194]], $MachinePrecision]], N[(x * N[(N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.00000000000000019e120 or 9.99999999999999945e193 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 84.4%

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

      1. unpow2 84.4%

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

   4. Simplified 84.4%

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

   if 5.00000000000000019e120 < (*.f64 y y) < 9.99999999999999945e193

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 11.8%

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

      1. unpow2 11.8%

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

   4. Simplified 11.8%

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

      1. distribute-rgt-in 11.8%

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

      2. *-un-lft-identity 11.8%

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

      3. flip-+ 65.3%

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

      4. associate-*l* 65.3%

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

      5. associate-*l* 65.3%

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

      6. associate-*l* 65.3%

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

   6. Applied egg-rr 65.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+120} \lor \neg \left(y \cdot y \leq 10^{+194}\right):\\ \;\;\;\;x \cdot \left(y \cdot y + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x - \left(y \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right)}{x - y \cdot \left(x \cdot y\right)}\\ \end{array} \]

Alternative 4: 81.6% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.05:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.05) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.05) {
		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) <= 0.05d0) 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) <= 0.05) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.05:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.05)
		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) <= 0.05)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.05], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. unpow2 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in y around 0 99.2%

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

   if 0.050000000000000003 < (*.f64 y y)

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 60.4%

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

      1. unpow2 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in y around inf 60.4%

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

      1. unpow2 60.4%

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

   7. Simplified 60.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.05:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5: 81.9% 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 80.4%

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

   1. unpow2 80.4%

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

4. Simplified 80.4%

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

5. Final simplification 80.4%

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

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 80.4%

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

   1. unpow2 80.4%

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

4. Simplified 80.4%

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

5. Taylor expanded in y around 0 52.5%

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

6. Final simplification 52.5%

   \[\leadsto x \]

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 2023275 
(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))))