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

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 5

Speedup: 1.0×

• 49.5% 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, 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 2: 80.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x \cdot y\right)\\ \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x + t_0\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\frac{t_0 \cdot t_0 - x \cdot x}{t_0 - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* x y))))
   (if (<= (* y y) 2e+54)
     (+ x t_0)
     (if (<= (* y y) 2e+262)
       (/ (- (* t_0 t_0) (* x x)) (- t_0 x))
       (* x (* y y))))))

C

double code(double x, double y) {
	double t_0 = y * (x * y);
	double tmp;
	if ((y * y) <= 2e+54) {
		tmp = x + t_0;
	} else if ((y * y) <= 2e+262) {
		tmp = ((t_0 * t_0) - (x * x)) / (t_0 - 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) :: t_0
    real(8) :: tmp
    t_0 = y * (x * y)
    if ((y * y) <= 2d+54) then
        tmp = x + t_0
    else if ((y * y) <= 2d+262) then
        tmp = ((t_0 * t_0) - (x * x)) / (t_0 - x)
    else
        tmp = x * (y * y)
    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) <= 2e+54) {
		tmp = x + t_0;
	} else if ((y * y) <= 2e+262) {
		tmp = ((t_0 * t_0) - (x * x)) / (t_0 - x);
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x * y)
	tmp = 0
	if (y * y) <= 2e+54:
		tmp = x + t_0
	elif (y * y) <= 2e+262:
		tmp = ((t_0 * t_0) - (x * x)) / (t_0 - x)
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x * y))
	tmp = 0.0
	if (Float64(y * y) <= 2e+54)
		tmp = Float64(x + t_0);
	elseif (Float64(y * y) <= 2e+262)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(t_0 - x));
	else
		tmp = Float64(x * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x * y);
	tmp = 0.0;
	if ((y * y) <= 2e+54)
		tmp = x + t_0;
	elseif ((y * y) <= 2e+262)
		tmp = ((t_0 * t_0) - (x * x)) / (t_0 - x);
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * y), $MachinePrecision], 2e+54], N[(x + t$95$0), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e+262], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2.0000000000000002e54

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 93.5%

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

      1. fma-def 93.5%

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, x, x\right)} \]

      2. unpow2 93.5%

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

   4. Simplified 93.5%

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

      1. fma-udef 93.5%

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

      2. associate-*l* 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 2.0000000000000002e54 < (*.f64 y y) < 2e262

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 21.7%

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

      1. fma-def 21.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, x, x\right)} \]

      2. unpow2 21.7%

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

   4. Simplified 21.7%

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

      1. fma-udef 21.7%

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

      2. flip-+ 45.3%

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

      3. associate-*l* 45.3%

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

      4. associate-*l* 45.3%

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

      5. associate-*l* 45.3%

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

   6. Applied egg-rr 45.3%

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

   if 2e262 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 95.3%

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

      1. fma-def 95.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, x, x\right)} \]

      2. unpow2 95.3%

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

   4. Simplified 95.3%

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

   5. Taylor expanded in y around inf 95.3%

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

      1. *-commutative 95.3%

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

      2. unpow2 95.3%

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

   7. Simplified 95.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x + y \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{+262}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot y\right)\right) \cdot \left(y \cdot \left(x \cdot y\right)\right) - x \cdot x}{y \cdot \left(x \cdot y\right) - x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 3: 81.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 0.0005) x (* x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 0.0005) {
		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) <= 0.0005d0) 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) <= 0.0005) {
		tmp = x;
	} else {
		tmp = x * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 0.0005:
		tmp = x
	else:
		tmp = x * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 0.0005)
		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) <= 0.0005)
		tmp = x;
	else
		tmp = x * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 0.0005], x, N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.1%

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

   if 5.0000000000000001e-4 < (*.f64 y y)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 68.7%

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

      1. fma-def 68.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, x, x\right)} \]

      2. unpow2 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in y around inf 68.7%

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

      1. *-commutative 68.7%

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

      2. unpow2 68.7%

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

   7. Simplified 68.7%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 0.0005:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4: 81.8% 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 84.6%

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

   1. fma-def 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, x, x\right)} \]

   2. unpow2 84.6%

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

4. Simplified 84.6%

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

   1. fma-udef 84.6%

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

   2. distribute-lft1-in 84.5%

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

6. Applied egg-rr 84.5%

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

7. Final simplification 84.5%

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

Alternative 5: 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 2023268 
(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))))