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

Percentage Accurate: 100.0% → 100.0%

Time: 1.6s

Alternatives: 6

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    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]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * (exp((y * y)))
END code

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    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]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * (exp((y * y)))
END code

Alternative 1: 100.0% accurate, 0.5× speedup

Math

\[x \cdot {\left(\sqrt{e^{\left|y\right|}}\right)}^{\left(\left|y\right| + \left|y\right|\right)} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x (pow (sqrt (exp (fabs y))) (+ (fabs y) (fabs y)))))

C

double code(double x, double y) {
	return x * pow(sqrt(exp(fabs(y))), (fabs(y) + fabs(y)));
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (sqrt(exp(abs(y))) ** (abs(y) + abs(y)))
end function

Java

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

Python

def code(x, y):
	return x * math.pow(math.sqrt(math.exp(math.fabs(y))), (math.fabs(y) + math.fabs(y)))

Julia

function code(x, y)
	return Float64(x * (sqrt(exp(abs(y))) ^ Float64(abs(y) + abs(y))))
end

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((sqrt((exp((abs(y)))))) ^ ((abs(y)) + (abs(y))))
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.6× speedup

Math

\[x \cdot {\left(e^{y}\right)}^{y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* 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)
use fmin_fmax_functions
    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]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((exp(y)) ^ y)
END code

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

Alternative 3: 89.5% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, x, x\right) \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fma (sqrt (* (* y y) (* y y))) x x))

C

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

Julia

function code(x, y)
	return fma(sqrt(Float64(Float64(y * y) * Float64(y * y))), x, x)
end

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((sqrt(((y * y) * (y * y)))) * x) + x
END code

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 81.8%

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

6. Applied rewrites 81.8%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 89.5%

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

8. Applied rewrites 89.5%

   \[\leadsto \mathsf{fma}\left(\sqrt{\left(y \cdot y\right) \cdot \left(y \cdot y\right)}, x, x\right) \]
9. Add Preprocessing

Alternative 4: 81.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (fma (* y y) x x))

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	((y * y) * x) + x
END code

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

4. Applied rewrites 81.8%

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

6. Applied rewrites 81.8%

   \[\leadsto \mathsf{fma}\left(y \cdot y, x, x\right) \]
7. Add Preprocessing

Alternative 5: 51.3% accurate, 4.5× speedup

Math

\[x \cdot 1 \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x 1.0))

C

double code(double x, double y) {
	return x * 1.0;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 1.0d0
end function

Java

public static double code(double x, double y) {
	return x * 1.0;
}

Python

def code(x, y):
	return x * 1.0

Julia

function code(x, y)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_] := N[(x * 1.0), $MachinePrecision]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * (1)
END code

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
3. Step-by-step derivation

4. Applied rewrites 51.3%

   \[\leadsto x \cdot 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Data.Number.Erf:$dmerfcx from erf-2.0.0.0"
  :precision binary64
  (* x (exp (* y y))))