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

Percentage Accurate: 99.8% → 99.9%

Time: 12.9s

Alternatives: 10

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. sub-neg 99.8%

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

   2. distribute-rgt-in 99.8%

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

   3. +-commutative 99.8%

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

   4. fma-undefine 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. fma-undefine 99.9%

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

   8. fma-undefine 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} \cdot 0.70711 - x \cdot 0.70711 \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.16)))
   (* x -0.70711)
   (* 0.70711 (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437))))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.16d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 0.70711d0 * (2.30753d0 + (x * ((x * 1.900161040244073d0) - 3.0191289437d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.16):
		tmp = x * -0.70711
	else:
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.16))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) - 3.0191289437))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.16)))
		tmp = x * -0.70711;
	else
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.16]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(0.70711 * N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] - 3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if -1.05000000000000004 < x < 1.15999999999999992

   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. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.4%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 3.0191289437\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * (x * (0.04481 + (0.99229 / x)))))) - 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[(x * N[(0.04481 + N[(0.99229 / x), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in x around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-define 99.9%

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

5. Simplified 99.9%

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

6. Taylor expanded in x around inf 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. un-div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

   2. +-commutative 99.9%

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

10. Simplified 99.9%

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

11. Final simplification 99.9%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.16)))
   (* x -0.70711)
   (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.16d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.16):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.16))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.16)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.16]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.15999999999999992 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if -1.05000000000000004 < x < 1.15999999999999992

   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 98.4%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \]
5. 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 99.8%

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

Alternative 6: 99.0% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (* x -0.70711)
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if -1.05000000000000004 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 97.8%

      \[\leadsto \color{blue}{1.6316775383 + -2.134856267379707 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 97.8%

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

   5. Simplified 97.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

5. Simplified 98.7%

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

Alternative 8: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.5) (not (<= x 1.16))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.5d0)) .or. (.not. (x <= 1.16d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.5) or not (x <= 1.16):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.5) || !(x <= 1.16))
		tmp = Float64(x * -0.70711);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.5) || ~((x <= 1.16)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.5], N[Not[LessEqual[x, 1.16]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.15999999999999992 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

   if -3.5 < x < 1.15999999999999992

   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 96.4%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.16\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.70711 * N[(N[(2.30753 / N[(1.0 + N[(x * 0.99229), $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. Add Preprocessing

3. Taylor expanded in x around 0 98.7%

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

   1. *-commutative 98.7%

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

5. Simplified 98.7%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{2.30753}}{1 + x \cdot 0.99229} - x\right) \]
7. Add Preprocessing

Alternative 10: 51.2% 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 99.8%

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

3. Taylor expanded in x around 0 48.2%

   \[\leadsto \color{blue}{1.6316775383} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(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)))