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

Percentage Accurate: 99.8% → 99.9%

Time: 13.4s

Alternatives: 13

Speedup: 1.0×

Specification

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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1.6316775383 + x \cdot 0.1913510371}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} + x \cdot -0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (/
   (+ 1.6316775383 (* x 0.1913510371))
   (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
  (* x -0.70711)))

C

double code(double x) {
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) + (x * -0.70711);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.6316775383d0 + (x * 0.1913510371d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) + (x * (-0.70711d0))
end function

Java

public static double code(double x) {
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) + (x * -0.70711);
}

Python

def code(x):
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) + (x * -0.70711)

Julia

function code(x)
	return Float64(Float64(Float64(1.6316775383 + Float64(x * 0.1913510371)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) + Float64(x * -0.70711))
end

MATLAB

function tmp = code(x)
	tmp = ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) + (x * -0.70711);
end

Wolfram

code[x_] := N[(N[(N[(1.6316775383 + N[(x * 0.1913510371), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} - \left(\frac{58.14938538768042}{x \cdot x} - x \cdot -0.70711\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (-
    (/ 4.2702753202410175 x)
    (- (/ 58.14938538768042 (* x x)) (* x -0.70711)))
   (if (<= x 1.15)
     (+ 1.6316775383 (* x (+ (* x 1.3436228731669864) -2.134856267379707)))
     (+ (/ (- 4.2702753202410175 (/ 58.14938538768042 x)) x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) - ((58.14938538768042 / (x * x)) - (x * -0.70711));
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (4.2702753202410175d0 / x) - ((58.14938538768042d0 / (x * x)) - (x * (-0.70711d0)))
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) + (-2.134856267379707d0)))
    else
        tmp = ((4.2702753202410175d0 - (58.14938538768042d0 / x)) / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) - ((58.14938538768042 / (x * x)) - (x * -0.70711));
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (4.2702753202410175 / x) - ((58.14938538768042 / (x * x)) - (x * -0.70711))
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707))
	else:
		tmp = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(4.2702753202410175 / x) - Float64(Float64(58.14938538768042 / Float64(x * x)) - Float64(x * -0.70711)));
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) + -2.134856267379707)));
	else
		tmp = Float64(Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 / x)) / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (4.2702753202410175 / x) - ((58.14938538768042 / (x * x)) - (x * -0.70711));
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	else
		tmp = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] - N[(N[(58.14938538768042 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(4.2702753202410175 - N[(58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.1499999999999999

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1.1499999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4.2702753202410175 - \frac{58.14938538768042}{x}}{x} + x \cdot -0.70711\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          (/ (- 4.2702753202410175 (/ 58.14938538768042 x)) x)
          (* x -0.70711))))
   (if (<= x -1.05)
     t_0
     (if (<= x 1.15)
       (+ 1.6316775383 (* x (+ (* x 1.3436228731669864) -2.134856267379707)))
       t_0))))

C

double code(double x) {
	double t_0 = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.2702753202410175d0 - (58.14938538768042d0 / x)) / x) + (x * (-0.70711d0))
    if (x <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) + (-2.134856267379707d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711)
	tmp = 0
	if x <= -1.05:
		tmp = t_0
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(4.2702753202410175 - Float64(58.14938538768042 / x)) / x) + Float64(x * -0.70711))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) + -2.134856267379707)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((4.2702753202410175 - (58.14938538768042 / x)) / x) + (x * -0.70711);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(4.2702753202410175 - N[(58.14938538768042 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.1499999999999999

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.6)
     (+ 1.6316775383 (* x (+ (* x 1.3436228731669864) -2.134856267379707)))
     (+ (/ 4.2702753202410175 x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.6) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 1.6d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) + (-2.134856267379707d0)))
    else
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.6) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 1.6:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707))
	else:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 1.6)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) + -2.134856267379707)));
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 1.6)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	else
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.6000000000000001

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 1.6000000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 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 6: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 0.75)
     (+ 1.6316775383 (* x -2.134856267379707))
     (+ (/ 4.2702753202410175 x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 0.75d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 0.75:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 0.75)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 0.75)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.75], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 0.75

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.75 < x

   1. Initial program 99.9%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.75:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.70711 (- (/ 6.039053782637804 x) x))))
   (if (<= x -1.05)
     t_0
     (if (<= x 0.75) (+ 1.6316775383 (* x -2.134856267379707)) t_0))))

C

double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    if (x <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 0.75d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 0.75) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.70711 * ((6.039053782637804 / x) - x)
	tmp = 0
	if x <= -1.05:
		tmp = t_0
	elif x <= 0.75:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x))
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 0.75)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.70711 * ((6.039053782637804 / x) - x);
	tmp = 0.0;
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 0.75)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 0.75], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.75 < x

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 0.75

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (/ x -1.4142071247754946)
   (if (<= x 1.15)
     (+ 1.6316775383 (* x -2.134856267379707))
     (/ x -1.4142071247754946))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x / (-1.4142071247754946d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = x / (-1.4142071247754946d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x / -1.4142071247754946
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = x / -1.4142071247754946
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x / -1.4142071247754946);
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(x / -1.4142071247754946);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x / -1.4142071247754946;
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = x / -1.4142071247754946;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x / -1.4142071247754946), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(x / -1.4142071247754946), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.1499999999999999

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{1.6316775383 + x \cdot 0.1913510371}{1 + x \cdot 0.99229} + x \cdot -0.70711 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (/ (+ 1.6316775383 (* x 0.1913510371)) (+ 1.0 (* x 0.99229)))
  (* x -0.70711)))

C

double code(double x) {
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * 0.99229))) + (x * -0.70711);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.6316775383d0 + (x * 0.1913510371d0)) / (1.0d0 + (x * 0.99229d0))) + (x * (-0.70711d0))
end function

Java

public static double code(double x) {
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * 0.99229))) + (x * -0.70711);
}

Python

def code(x):
	return ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * 0.99229))) + (x * -0.70711)

Julia

function code(x)
	return Float64(Float64(Float64(1.6316775383 + Float64(x * 0.1913510371)) / Float64(1.0 + Float64(x * 0.99229))) + Float64(x * -0.70711))
end

MATLAB

function tmp = code(x)
	tmp = ((1.6316775383 + (x * 0.1913510371)) / (1.0 + (x * 0.99229))) + (x * -0.70711);
end

Wolfram

code[x_] := N[(N[(N[(1.6316775383 + N[(x * 0.1913510371), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Alternative 10: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-1.4142071247754946}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (/ x -1.4142071247754946)
   (if (<= x 1.15) 1.6316775383 (/ x -1.4142071247754946))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x / (-1.4142071247754946d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0
    else
        tmp = x / (-1.4142071247754946d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x / -1.4142071247754946;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x / -1.4142071247754946;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x / -1.4142071247754946
	elif x <= 1.15:
		tmp = 1.6316775383
	else:
		tmp = x / -1.4142071247754946
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x / -1.4142071247754946);
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = Float64(x / -1.4142071247754946);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x / -1.4142071247754946;
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = x / -1.4142071247754946;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x / -1.4142071247754946), $MachinePrecision], If[LessEqual[x, 1.15], 1.6316775383, N[(x / -1.4142071247754946), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.1499999999999999

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (* x -0.70711) (if (<= x 1.15) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   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 inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if -1.05000000000000004 < x < 1.1499999999999999

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

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 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

Alternative 13: 9.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.1928378166664987 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.1928378166664987)

C

double code(double x) {
	return 0.1928378166664987;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.1928378166664987d0
end function

Java

public static double code(double x) {
	return 0.1928378166664987;
}

Python

def code(x):
	return 0.1928378166664987

Julia

function code(x)
	return 0.1928378166664987
end

MATLAB

function tmp = code(x)
	tmp = 0.1928378166664987;
end

Wolfram

code[x_] := 0.1928378166664987

Derivation

1. Initial program 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in x around inf 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

11. Taylor expanded in x around 0 0

    \[\leadsto expr\]

13. Simplified 0

    \[\leadsto expr\]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, B"
  :precision binary64
  (* 0.70711 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x)))