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

Percentage Accurate: 100.0% → 100.0%

Time: 26.9s

Alternatives: 21

Speedup: 1.0×

• 50.0% 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

\[\begin{array}{l} \\ e^{y \cdot y} \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (exp (* y y)) x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot y\\ \mathbf{if}\;e^{y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, \mathsf{fma}\left(t\_0, 0.5, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(t\_0 \cdot y\right) \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) y)))
   (if (<= (exp (* y y)) 2.0)
     (fma (* y x) (fma t_0 0.5 y) x)
     (* (* (* (* (* t_0 y) y) y) 0.16666666666666666) x))))

C

double code(double x, double y) {
	double t_0 = (y * y) * y;
	double tmp;
	if (exp((y * y)) <= 2.0) {
		tmp = fma((y * x), fma(t_0, 0.5, y), x);
	} else {
		tmp = ((((t_0 * y) * y) * y) * 0.16666666666666666) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * y)
	tmp = 0.0
	if (exp(Float64(y * y)) <= 2.0)
		tmp = fma(Float64(y * x), fma(t_0, 0.5, y), x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(t_0 * y) * y) * y) * 0.16666666666666666) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[Exp[N[(y * y), $MachinePrecision]], $MachinePrecision], 2.0], N[(N[(y * x), $MachinePrecision] * N[(t$95$0 * 0.5 + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(N[(t$95$0 * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \cdot y} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), y, 1\right)} \]

   5. Applied rewrites 99.8%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot y, \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 0.5\right), y\right), y, 1\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-lft-identity N/A

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

      13. distribute-rgt-in N/A

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

      14. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot x, \mathsf{fma}\left(\left(y \cdot y\right) \cdot y, 0.5, y\right), x\right)} \]

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({y}^{2} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right)\right) \cdot y} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {y}^{2}\right)\right), y, 1\right)} \]

   5. Applied rewrites 88.7%

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot y, \mathsf{fma}\left(0.16666666666666666 \cdot y, y, 0.5\right), y\right), y, 1\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x \cdot \left(\frac{1}{6} \cdot \color{blue}{{y}^{6}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 88.7%

      \[\leadsto x \cdot \left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot \color{blue}{0.16666666666666666}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y \cdot y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(y \cdot x, \mathsf{fma}\left(\left(y \cdot y\right) \cdot y, 0.5, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(\left(\left(y \cdot y\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right) \cdot x\\ \end{array} \]
5. 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 2024230 
(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))))