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

Percentage Accurate: 99.8% → 99.8%

Time: 9.9s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. +-commutative 99.9%

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

   6. +-commutative 99.9%

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

   7. fma-udef 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-udef 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. Step-by-step derivation

   1. fma-udef 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-udef 99.9%

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

8. Applied egg-rr 99.9%

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

9. Taylor expanded in x around 0 99.9%

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

    1. *-commutative 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;\left(1.6316775383 + x \cdot -1.427746267379707\right) - x \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 0.74)
     (- (+ 1.6316775383 (* x -1.427746267379707)) (* x 0.70711))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 0.74) {
		tmp = (1.6316775383 + (x * -1.427746267379707)) - (x * 0.70711);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	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 <= 0.74d0) then
        tmp = (1.6316775383d0 + (x * (-1.427746267379707d0))) - (x * 0.70711d0)
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    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 <= 0.74) {
		tmp = (1.6316775383 + (x * -1.427746267379707)) - (x * 0.70711);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 0.74:
		tmp = (1.6316775383 + (x * -1.427746267379707)) - (x * 0.70711)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 0.74)
		tmp = Float64(Float64(1.6316775383 + Float64(x * -1.427746267379707)) - Float64(x * 0.70711));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 0.74)
		tmp = (1.6316775383 + (x * -1.427746267379707)) - (x * 0.70711);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 0.74], N[(N[(1.6316775383 + N[(x * -1.427746267379707), $MachinePrecision]), $MachinePrecision] - N[(x * 0.70711), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 < 0.73999999999999999

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-udef 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-udef 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. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{\left(1.6316775383 + -1.427746267379707 \cdot x\right)} + \left(-x\right) \cdot 0.70711 \]
   6. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \left(1.6316775383 + \color{blue}{x \cdot -1.427746267379707}\right) + \left(-x\right) \cdot 0.70711 \]

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\left(1.6316775383 + x \cdot -1.427746267379707\right)} + \left(-x\right) \cdot 0.70711 \]

   if 0.73999999999999999 < 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 99.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;\left(1.6316775383 + x \cdot -1.427746267379707\right) - x \cdot 0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $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. Final simplification 99.9%

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

Alternative 4: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 0.74)
     (+ 1.6316775383 (* x -2.134856267379707))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 0.74) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	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 <= 0.74d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    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 <= 0.74) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 0.74:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 0.74)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 0.74)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 0.74], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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 < 0.73999999999999999

   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 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

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

   if 0.73999999999999999 < 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 99.5%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% 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 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   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 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]
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 6: 98.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(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 <= (-1.05d0)) .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 <= -1.05) || !(x <= 1.16)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	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
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(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 <= -1.05) || ~((x <= 1.16)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	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], 1.6316775383]

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 99.5%

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

      1. *-commutative 99.5%

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

   5. Simplified 99.5%

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

   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 96.8%

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

4. Final simplification 98.0%

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

Alternative 7: 50.4% 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. Add Preprocessing

3. Taylor expanded in x around 0 54.0%

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

4. Final simplification 54.0%

   \[\leadsto 1.6316775383 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(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)))