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

Percentage Accurate: 99.9% → 99.8%

Time: 7.0s

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: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right) - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (expm1
    (log1p (/ (fma x 0.27061 2.30753) (fma x (fma x 0.04481 0.99229) 1.0))))
   x)))

C

double code(double x) {
	return 0.70711 * (expm1(log1p((fma(x, 0.27061, 2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)))) - x);
}

Julia

function code(x)
	return Float64(0.70711 * Float64(expm1(log1p(Float64(fma(x, 0.27061, 2.30753) / fma(x, fma(x, 0.04481, 0.99229), 1.0)))) - x))
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(Exp[N[Log[1 + N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(x * N[(x * 0.04481 + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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. Step-by-step derivation

   1. expm1-log1p-u 99.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x\right) \]

   2. +-commutative 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

   3. fma-def 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

   4. +-commutative 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right)\right) - x\right) \]

   5. fma-def 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right)\right) - x\right) \]

   6. +-commutative 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)}\right)\right) - x\right) \]

   7. fma-def 99.9%

      \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)}\right)\right) - x\right) \]

3. Applied egg-rr 99.9%

   \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right)} - x\right) \]

4. Final simplification 99.9%

   \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right) - x\right) \]

Alternative 2: 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{58.14938538768042}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.4)
   (+
    (/ 4.2702753202410175 x)
    (- (* x -0.70711) (/ 58.14938538768042 (* x x))))
   (if (<= x 1.15) (+ (* x -2.134856267379707) 1.6316775383) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -5.4) {
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	} 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 <= (-5.4d0)) then
        tmp = (4.2702753202410175d0 / x) + ((x * (-0.70711d0)) - (58.14938538768042d0 / (x * x)))
    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 <= -5.4) {
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	} 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 <= -5.4:
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)))
	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 <= -5.4)
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(Float64(x * -0.70711) - Float64(58.14938538768042 / Float64(x * x))));
	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 <= -5.4)
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	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, -5.4], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(N[(x * -0.70711), $MachinePrecision] - N[(58.14938538768042 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if x < -5.4000000000000004

   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 98.6%

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

      1. associate--l+ 98.6%

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

      2. associate-*r/ 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r/ 98.6%

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

      6. metadata-eval 98.6%

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

      7. unpow2 98.6%

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

   4. Simplified 98.6%

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

   if -5.4000000000000004 < 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. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{-2.134856267379707 \cdot x + 1.6316775383} \]

   if 1.1499999999999999 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{58.14938538768042}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 4: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		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 <= (-1.05d0)) 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 <= -1.05) {
		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 <= -1.05:
		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 <= -1.05)
		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 <= -1.05)
		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, -1.05], 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 < -1.05000000000000004 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 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   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. Taylor expanded in x around 0 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 5: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (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 <= (-1.05d0)) then
        tmp = (4.2702753202410175d0 / x) + (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 <= -1.05) {
		tmp = (4.2702753202410175 / x) + (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 <= -1.05:
		tmp = (4.2702753202410175 / x) + (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 <= -1.05)
		tmp = Float64(Float64(4.2702753202410175 / x) + 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 <= -1.05)
		tmp = (4.2702753202410175 / x) + (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, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $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 3 regimes

2. if x < -1.05000000000000004

   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 98.5%

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

      1. +-commutative 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

   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. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{-2.134856267379707 \cdot x + 1.6316775383} \]

   if 1.1499999999999999 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 6: 98.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (* x -0.70711) (if (<= x 1.2) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		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 <= 1.2:
		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 <= 1.2)
		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 <= 1.2)
		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, 1.2], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < 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 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   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. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

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 99.9%

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

2. Taylor expanded in x around 0 47.5%

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

3. Final simplification 47.5%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023196 
(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)))