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

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

• 49.9% 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

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * pow(exp((2.0 * y_m)), (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((2.0d0 * y_m)) ** (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((2.0 * y_m)), (y_m * 0.5));
}

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[Power[N[Exp[N[(2.0 * y$95$m), $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-prod-down 100.0%

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

   4. add-exp-log 100.0%

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

   5. pow2 100.0%

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

   6. log-pow 100.0%

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

   7. add-log-exp 100.0%

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

   8. div-inv 100.0%

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

   9. metadata-eval 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x + x \cdot \mathsf{expm1}\left(y\_m \cdot y\_m\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (+ x (* x (expm1 (* y_m y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x + (x * expm1((y_m * y_m)));
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x + (x * Math.expm1((y_m * y_m)));
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x + (x * math.expm1((y_m * y_m)))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x + Float64(x * expm1(Float64(y_m * y_m))))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x + N[(x * N[(Exp[N[(y$95$m * y$95$m), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y \cdot y\right)\right)}} \]

   2. log1p-undefine 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(y \cdot y\right)\right)}} \]

   3. add-exp-log 100.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + \mathsf{expm1}\left(y \cdot y\right)\right)} \]

   4. pow2 100.0%

      \[\leadsto x \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{{y}^{2}}\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left(1 + \mathsf{expm1}\left({y}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{expm1}\left({y}^{2}\right) + 1\right)} \]

   2. distribute-lft-in 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{expm1}\left({y}^{2}\right) + x \cdot 1} \]

   3. *-rgt-identity 100.0%

      \[\leadsto x \cdot \mathsf{expm1}\left({y}^{2}\right) + \color{blue}{x} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{expm1}\left({y}^{2}\right) + x} \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \mathsf{expm1}\left(\color{blue}{y \cdot y}\right) + x \]

8. Applied egg-rr 100.0%

   \[\leadsto x \cdot \mathsf{expm1}\left(\color{blue}{y \cdot y}\right) + x \]

9. Final simplification 100.0%

   \[\leadsto x + x \cdot \mathsf{expm1}\left(y \cdot y\right) \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x \cdot \left(\mathsf{expm1}\left(y\_m \cdot y\_m\right) + 1\right) \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (* x (+ (expm1 (* y_m y_m)) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x * (expm1((y_m * y_m)) + 1.0);
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return x * (Math.expm1((y_m * y_m)) + 1.0);
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return x * (math.expm1((y_m * y_m)) + 1.0)

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(x * Float64(expm1(Float64(y_m * y_m)) + 1.0))
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x * N[(N[(Exp[N[(y$95$m * y$95$m), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. log1p-expm1-u 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y \cdot y\right)\right)}} \]

   2. log1p-undefine 100.0%

      \[\leadsto x \cdot e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(y \cdot y\right)\right)}} \]

   3. add-exp-log 100.0%

      \[\leadsto x \cdot \color{blue}{\left(1 + \mathsf{expm1}\left(y \cdot y\right)\right)} \]

   4. pow2 100.0%

      \[\leadsto x \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{{y}^{2}}\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto x \cdot \color{blue}{\left(1 + \mathsf{expm1}\left({y}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto x \cdot \mathsf{expm1}\left(\color{blue}{y \cdot y}\right) + x \]

6. Applied egg-rr 100.0%

   \[\leadsto x \cdot \left(1 + \mathsf{expm1}\left(\color{blue}{y \cdot y}\right)\right) \]

7. Final simplification 100.0%

   \[\leadsto x \cdot \left(\mathsf{expm1}\left(y \cdot y\right) + 1\right) \]
8. Add Preprocessing

Alternative 4: 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. Add Preprocessing

Alternative 5: 81.9% accurate, 15.0× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m) {
	return x + (x * (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 + (x * (y_m * y_m))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(x + N[(x * 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. Taylor expanded in y around 0 83.3%

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

   1. *-commutative 83.3%

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

5. Simplified 83.3%

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

   1. unpow2 100.0%

      \[\leadsto x \cdot \mathsf{expm1}\left(\color{blue}{y \cdot y}\right) + x \]

7. Applied egg-rr 83.3%

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

8. Final simplification 83.3%

   \[\leadsto x + x \cdot \left(y \cdot y\right) \]
9. Add Preprocessing

Alternative 6: 51.3% 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 51.3%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024123 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64

  :alt
  (! :herbie-platform default (* x (pow (exp y) y)))

  (* x (exp (* y y))))