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

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 1.0×

• 49.1% 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

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot {\left(e^{y_m \cdot 2}\right)}^{\left(y_m \cdot 0.5\right)} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (pow (exp (* y_m 2.0)) (* y_m 0.5))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * pow(exp((y_m * 2.0)), (y_m * 0.5));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = x * (exp((y_m * 2.0d0)) ** (y_m * 0.5d0))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x * Math.pow(Math.exp((y_m * 2.0)), (y_m * 0.5));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x * math.pow(math.exp((y_m * 2.0)), (y_m * 0.5))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * (exp(Float64(y_m * 2.0)) ^ Float64(y_m * 0.5)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = x * (exp((y_m * 2.0)) ^ (y_m * 0.5));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[Power[N[Exp[N[(y$95$m * 2.0), $MachinePrecision]], $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing
3. 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-sqr 100.0%

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

   4. pow-pow 100.0%

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

   5. pow-exp 100.0%

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

   6. div-inv 100.0%

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

   7. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

   \[\leadsto x \cdot {\left(e^{y \cdot 2}\right)}^{\left(y \cdot 0.5\right)} \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 0.5× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (pow (exp y_m) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * pow(exp(y_m), y_m);
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = x * (exp(y_m) ** y_m)
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x * Math.pow(Math.exp(y_m), y_m);
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x * math.pow(math.exp(y_m), y_m)

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * (exp(y_m) ^ y_m))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = x * (exp(y_m) ^ y_m);
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[Power[N[Exp[y$95$m], $MachinePrecision], y$95$m], $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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto x \cdot {\left(e^{y}\right)}^{y} \]
6. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot e^{y_m \cdot y_m} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (exp (* y_m y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * exp((y_m * y_m));
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = x * exp((y_m * y_m))
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x * Math.exp((y_m * y_m));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x * math.exp((y_m * y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * exp(Float64(y_m * y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = x * exp((y_m * y_m));
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[Exp[N[(y$95$m * y$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto x \cdot e^{y \cdot y} \]
4. Add Preprocessing

Alternative 4: 51.8% accurate, 105.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 x)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = x
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x

Julia

y_m = abs(y)
function code(x, y_m)
	return x
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = x;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := x

Derivation

1. Initial program 100.0%

   \[x \cdot e^{y \cdot y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 57.6%

   \[\leadsto x \cdot \color{blue}{1} \]

4. Final simplification 57.6%

   \[\leadsto x \]
5. Add Preprocessing

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 2024013 
(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))))