Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B

Percentage Accurate: 99.8% → 99.8%

Time: 2.3s

Alternatives: 6

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * (((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Python

def code(x):
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 58.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.04481 - x\\ t_1 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\\ t_2 := t\_1 - x\\ \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 4:\\ \;\;\;\;0.70711 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x 0.04481) x))
        (t_1
         (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))))
        (t_2 (- t_1 x)))
   (if (<= t_2 -50.0) t_0 (if (<= t_2 4.0) (* 0.70711 t_1) t_0))))

C

double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double t_1 = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	double t_2 = t_1 - x;
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_0;
	} else if (t_2 <= 4.0) {
		tmp = 0.70711 * t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * 0.04481d0) - x
    t_1 = (2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))
    t_2 = t_1 - x
    if (t_2 <= (-50.0d0)) then
        tmp = t_0
    else if (t_2 <= 4.0d0) then
        tmp = 0.70711d0 * t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double t_1 = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	double t_2 = t_1 - x;
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_0;
	} else if (t_2 <= 4.0) {
		tmp = 0.70711 * t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * 0.04481) - x
	t_1 = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))
	t_2 = t_1 - x
	tmp = 0
	if t_2 <= -50.0:
		tmp = t_0
	elif t_2 <= 4.0:
		tmp = 0.70711 * t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * 0.04481) - x)
	t_1 = Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481)))))
	t_2 = Float64(t_1 - x)
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = t_0;
	elseif (t_2 <= 4.0)
		tmp = Float64(0.70711 * t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * 0.04481) - x;
	t_1 = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	t_2 = t_1 - x;
	tmp = 0.0;
	if (t_2 <= -50.0)
		tmp = t_0;
	elseif (t_2 <= 4.0)
		tmp = 0.70711 * t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.04481), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - x), $MachinePrecision]}, If[LessEqual[t$95$2, -50.0], t$95$0, If[LessEqual[t$95$2, 4.0], N[(0.70711 * t$95$1), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -50 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]

   5. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right) - x \]

   8. Applied rewrites 20.7%

      \[\leadsto x \cdot 0.04481 - x \]

   if -50 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 100.0%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \color{blue}{\left(\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x\right)} \]

   5. Applied rewrites 3.6%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(x \cdot 0.04481\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{70711}{100000} \cdot \left(\frac{4481}{100000} \cdot \color{blue}{x}\right) \]

   8. Applied rewrites 99.4%

      \[\leadsto 0.70711 \cdot \frac{2.30753 + x \cdot 0.27061}{\color{blue}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 20.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.04481 - x\\ t_1 := 2.30753 + x \cdot 0.27061\\ t_2 := \frac{t\_1}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x 0.04481) x))
        (t_1 (+ 2.30753 (* x 0.27061)))
        (t_2 (- (/ t_1 (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))
   (if (<= t_2 -50.0) t_0 (if (<= t_2 4.0) t_1 t_0))))

C

double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double t_1 = 2.30753 + (x * 0.27061);
	double t_2 = (t_1 / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_0;
	} else if (t_2 <= 4.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (x * 0.04481d0) - x
    t_1 = 2.30753d0 + (x * 0.27061d0)
    t_2 = (t_1 / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if (t_2 <= (-50.0d0)) then
        tmp = t_0
    else if (t_2 <= 4.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double t_1 = 2.30753 + (x * 0.27061);
	double t_2 = (t_1 / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_0;
	} else if (t_2 <= 4.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * 0.04481) - x
	t_1 = 2.30753 + (x * 0.27061)
	t_2 = (t_1 / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if t_2 <= -50.0:
		tmp = t_0
	elif t_2 <= 4.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * 0.04481) - x)
	t_1 = Float64(2.30753 + Float64(x * 0.27061))
	t_2 = Float64(Float64(t_1 / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = t_0;
	elseif (t_2 <= 4.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * 0.04481) - x;
	t_1 = 2.30753 + (x * 0.27061);
	t_2 = (t_1 / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if (t_2 <= -50.0)
		tmp = t_0;
	elseif (t_2 <= 4.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.04481), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[t$95$2, -50.0], t$95$0, If[LessEqual[t$95$2, 4.0], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -50 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x)

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]

   5. Applied rewrites 20.3%

      \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right) - x \]

   8. Applied rewrites 20.7%

      \[\leadsto x \cdot 0.04481 - x \]

   if -50 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   1. Initial program 100.0%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

   5. Applied rewrites 20.2%

      \[\leadsto \color{blue}{2.30753 + x \cdot 0.27061} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 20.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]

5. Applied rewrites 20.3%

   \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x} \]
6. Add Preprocessing

Alternative 5: 10.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ 2.30753 + x \cdot 0.27061 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 2.30753 (* x 0.27061)))

C

double code(double x) {
	return 2.30753 + (x * 0.27061);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 + (x * 0.27061d0)
end function

Java

public static double code(double x) {
	return 2.30753 + (x * 0.27061);
}

Python

def code(x):
	return 2.30753 + (x * 0.27061)

Julia

function code(x)
	return Float64(2.30753 + Float64(x * 0.27061))
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 + (x * 0.27061);
end

Wolfram

code[x_] := N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + x \cdot \left(x \cdot \left(\frac{134362287316698645903}{100000000000000000000} + \frac{-12692862305735843227608787}{10000000000000000000000000} \cdot x\right) - \frac{2134856267379707}{1000000000000000}\right)} \]

5. Applied rewrites 10.6%

   \[\leadsto \color{blue}{2.30753 + x \cdot 0.27061} \]
6. Add Preprocessing

Alternative 6: 2.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.27061 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.27061))

C

double code(double x) {
	return x * 0.27061;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.27061d0
end function

Java

public static double code(double x) {
	return x * 0.27061;
}

Python

def code(x):
	return x * 0.27061

Julia

function code(x)
	return Float64(x * 0.27061)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.27061;
end

Wolfram

code[x_] := N[(x * 0.27061), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{16316775383}{10000000000} + \frac{-2134856267379707}{1000000000000000} \cdot x} \]

5. Applied rewrites 20.3%

   \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{230753}{100000} + \color{blue}{\frac{-30191289437}{10000000000} \cdot x} \]

8. Applied rewrites 2.2%

   \[\leadsto x \cdot \color{blue}{0.27061} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  :pre (TRUE)
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))