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

Percentage Accurate: 99.9% → 99.9%

Time: 6.2s

Alternatives: 7

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: 99.9% 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

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

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 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;t_0 - \frac{58.14938538768042}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\frac{0.70711}{x \cdot 0.5670050797822634 + 0.4333638132548656}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (/ 4.2702753202410175 x) (* x -0.70711))))
   (if (<= x -4.6)
     (- t_0 (/ 58.14938538768042 (* x x)))
     (if (<= x 2.2)
       (/ 0.70711 (+ (* x 0.5670050797822634) 0.4333638132548656))
       t_0))))

C

double code(double x) {
	double t_0 = (4.2702753202410175 / x) + (x * -0.70711);
	double tmp;
	if (x <= -4.6) {
		tmp = t_0 - (58.14938538768042 / (x * x));
	} else if (x <= 2.2) {
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	} 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 / x) + (x * (-0.70711d0))
    if (x <= (-4.6d0)) then
        tmp = t_0 - (58.14938538768042d0 / (x * x))
    else if (x <= 2.2d0) then
        tmp = 0.70711d0 / ((x * 0.5670050797822634d0) + 0.4333638132548656d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (4.2702753202410175 / x) + (x * -0.70711);
	double tmp;
	if (x <= -4.6) {
		tmp = t_0 - (58.14938538768042 / (x * x));
	} else if (x <= 2.2) {
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (4.2702753202410175 / x) + (x * -0.70711)
	tmp = 0
	if x <= -4.6:
		tmp = t_0 - (58.14938538768042 / (x * x))
	elif x <= 2.2:
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711))
	tmp = 0.0
	if (x <= -4.6)
		tmp = Float64(t_0 - Float64(58.14938538768042 / Float64(x * x)));
	elseif (x <= 2.2)
		tmp = Float64(0.70711 / Float64(Float64(x * 0.5670050797822634) + 0.4333638132548656));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (4.2702753202410175 / x) + (x * -0.70711);
	tmp = 0.0;
	if (x <= -4.6)
		tmp = t_0 - (58.14938538768042 / (x * x));
	elseif (x <= 2.2)
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6], N[(t$95$0 - N[(58.14938538768042 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2], N[(0.70711 / N[(N[(x * 0.5670050797822634), $MachinePrecision] + 0.4333638132548656), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5999999999999996

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{\left(4.2702753202410175 \cdot \frac{1}{x} + -0.70711 \cdot x\right) - 58.14938538768042 \cdot \frac{1}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \left(\color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x\right) - 58.14938538768042 \cdot \frac{1}{{x}^{2}} \]

      2. metadata-eval 98.9%

         \[\leadsto \left(\frac{\color{blue}{4.2702753202410175}}{x} + -0.70711 \cdot x\right) - 58.14938538768042 \cdot \frac{1}{{x}^{2}} \]

      3. *-commutative 98.9%

         \[\leadsto \left(\frac{4.2702753202410175}{x} + \color{blue}{x \cdot -0.70711}\right) - 58.14938538768042 \cdot \frac{1}{{x}^{2}} \]

      4. associate-*r/ 98.9%

         \[\leadsto \left(\frac{4.2702753202410175}{x} + x \cdot -0.70711\right) - \color{blue}{\frac{58.14938538768042 \cdot 1}{{x}^{2}}} \]

      5. metadata-eval 98.9%

         \[\leadsto \left(\frac{4.2702753202410175}{x} + x \cdot -0.70711\right) - \frac{\color{blue}{58.14938538768042}}{{x}^{2}} \]

      6. unpow2 98.9%

         \[\leadsto \left(\frac{4.2702753202410175}{x} + x \cdot -0.70711\right) - \frac{58.14938538768042}{\color{blue}{x \cdot x}} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\left(\frac{4.2702753202410175}{x} + x \cdot -0.70711\right) - \frac{58.14938538768042}{x \cdot x}} \]

   if -4.5999999999999996 < x < 2.2000000000000002

   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. Step-by-step derivation

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.70711 \cdot \left({\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x\right)}{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{0.70711}{\frac{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x}}} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{0.70711}{\frac{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x}}} \]

   6. Taylor expanded in x around 0 97.7%

      \[\leadsto \frac{0.70711}{\color{blue}{0.5670050797822634 \cdot x + 0.4333638132548656}} \]

   if 2.2000000000000002 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.7%

      \[\leadsto \color{blue}{4.2702753202410175 \cdot \frac{1}{x} + -0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x \]

      2. metadata-eval 99.7%

         \[\leadsto \frac{\color{blue}{4.2702753202410175}}{x} + -0.70711 \cdot x \]

      3. *-commutative 99.7%

         \[\leadsto \frac{4.2702753202410175}{x} + \color{blue}{x \cdot -0.70711} \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x} + x \cdot -0.70711} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6:\\ \;\;\;\;\left(\frac{4.2702753202410175}{x} + x \cdot -0.70711\right) - \frac{58.14938538768042}{x \cdot x}\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\frac{0.70711}{x \cdot 0.5670050797822634 + 0.4333638132548656}\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.55 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.55) (not (<= x 2.8)))
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (+ (* x -2.134856267379707) 1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.55) || !(x <= 2.8)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.55d0)) .or. (.not. (x <= 2.8d0))) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.55) || !(x <= 2.8)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.55) or not (x <= 2.8):
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	else:
		tmp = (x * -2.134856267379707) + 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.55) || !(x <= 2.8))
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	else
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.55) || ~((x <= 2.8)))
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	else
		tmp = (x * -2.134856267379707) + 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.55], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5499999999999998 or 2.7999999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{4.2702753202410175 \cdot \frac{1}{x} + -0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.3%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x \]

      2. metadata-eval 99.3%

         \[\leadsto \frac{\color{blue}{4.2702753202410175}}{x} + -0.70711 \cdot x \]

      3. *-commutative 99.3%

         \[\leadsto \frac{4.2702753202410175}{x} + \color{blue}{x \cdot -0.70711} \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x} + x \cdot -0.70711} \]

   if -2.5499999999999998 < x < 2.7999999999999998

   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. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{-2.134856267379707 \cdot x + 1.6316775383} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.55 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78 \lor \neg \left(x \leq 2.2\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;\frac{0.70711}{x \cdot 0.5670050797822634 + 0.4333638132548656}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.78) (not (<= x 2.2)))
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (/ 0.70711 (+ (* x 0.5670050797822634) 0.4333638132548656))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.78) || !(x <= 2.2)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.78d0)) .or. (.not. (x <= 2.2d0))) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else
        tmp = 0.70711d0 / ((x * 0.5670050797822634d0) + 0.4333638132548656d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.78) || !(x <= 2.2)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.78) or not (x <= 2.2):
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	else:
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.78) || !(x <= 2.2))
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	else
		tmp = Float64(0.70711 / Float64(Float64(x * 0.5670050797822634) + 0.4333638132548656));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.78) || ~((x <= 2.2)))
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	else
		tmp = 0.70711 / ((x * 0.5670050797822634) + 0.4333638132548656);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.78], N[Not[LessEqual[x, 2.2]], $MachinePrecision]], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(0.70711 / N[(N[(x * 0.5670050797822634), $MachinePrecision] + 0.4333638132548656), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.78000000000000003 or 2.2000000000000002 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.3%

      \[\leadsto \color{blue}{4.2702753202410175 \cdot \frac{1}{x} + -0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.3%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + -0.70711 \cdot x \]

      2. metadata-eval 99.3%

         \[\leadsto \frac{\color{blue}{4.2702753202410175}}{x} + -0.70711 \cdot x \]

      3. *-commutative 99.3%

         \[\leadsto \frac{4.2702753202410175}{x} + \color{blue}{x \cdot -0.70711} \]

   4. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x} + x \cdot -0.70711} \]

   if -0.78000000000000003 < x < 2.2000000000000002

   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. Step-by-step derivation

      1. flip-- 99.9%

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

      2. associate-*r/ 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.70711 \cdot \left({\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x\right)}{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]
   4. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\frac{0.70711}{\frac{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x}}} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{0.70711}{\frac{x + \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}{{\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2} - x \cdot x}}} \]

   6. Taylor expanded in x around 0 97.7%

      \[\leadsto \frac{0.70711}{\color{blue}{0.5670050797822634 \cdot x + 0.4333638132548656}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78 \lor \neg \left(x \leq 2.2\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;\frac{0.70711}{x \cdot 0.5670050797822634 + 0.4333638132548656}\\ \end{array} \]

Alternative 5: 98.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (* x -0.70711)
   (if (<= x 1.15) (+ (* x -2.134856267379707) 1.6316775383) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = (x * -2.134856267379707) + 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 <= (-1.06d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = (x * -2.134856267379707) + 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.06)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.06)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = (x * -2.134856267379707) + 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.06], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0600000000000001 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{x \cdot -0.70711} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{x \cdot -0.70711} \]

   if -1.0600000000000001 < x < 1.1499999999999999

   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. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{-2.134856267379707 \cdot x + 1.6316775383} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 6: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.45) (* x -0.70711) (if (<= x 1.15) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.45) {
		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 <= (-3.45d0)) 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 <= -3.45) {
		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 <= -3.45:
		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 <= -3.45)
		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 <= -3.45)
		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, -3.45], 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 < -3.4500000000000002 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.9%

      \[\leadsto \color{blue}{-0.70711 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{x \cdot -0.70711} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{x \cdot -0.70711} \]

   if -3.4500000000000002 < x < 1.1499999999999999

   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. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{1.6316775383} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.45:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 7: 51.4% 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 99.8%

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

2. Taylor expanded in x around 0 46.7%

   \[\leadsto \color{blue}{1.6316775383} \]

3. Final simplification 46.7%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023257 
(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)))