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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% 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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 2.0191289437\right)\right) - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -7.2) (not (<= x 6.2)))
   (*
    0.70711
    (-
     (/
      (+
       6.039053782637804
       (/ (+ (/ 1686.279566230464 x) -82.23527511657367) x))
      x)
     x))
   (*
    0.70711
    (-
     (+
      2.30753
      (*
       x
       (- (* x (+ 1.900161040244073 (* x -1.7950336306565942))) 2.0191289437)))
     x))))

C

double code(double x) {
	double tmp;
	if ((x <= -7.2) || !(x <= 6.2)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 2.0191289437))) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.2d0)) .or. (.not. (x <= 6.2d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 + (((1686.279566230464d0 / x) + (-82.23527511657367d0)) / x)) / x) - x)
    else
        tmp = 0.70711d0 * ((2.30753d0 + (x * ((x * (1.900161040244073d0 + (x * (-1.7950336306565942d0)))) - 2.0191289437d0))) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -7.2) || !(x <= 6.2)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 0.70711 * ((2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 2.0191289437))) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -7.2) or not (x <= 6.2):
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x)
	else:
		tmp = 0.70711 * ((2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 2.0191289437))) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -7.2) || !(x <= 6.2))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 + Float64(Float64(Float64(1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x));
	else
		tmp = Float64(0.70711 * Float64(Float64(2.30753 + Float64(x * Float64(Float64(x * Float64(1.900161040244073 + Float64(x * -1.7950336306565942))) - 2.0191289437))) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -7.2) || ~((x <= 6.2)))
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	else
		tmp = 0.70711 * ((2.30753 + (x * ((x * (1.900161040244073 + (x * -1.7950336306565942))) - 2.0191289437))) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -7.2], N[Not[LessEqual[x, 6.2]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 + N[(N[(N[(1686.279566230464 / x), $MachinePrecision] + -82.23527511657367), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(2.30753 + N[(x * N[(N[(x * N[(1.900161040244073 + N[(x * -1.7950336306565942), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000018 or 6.20000000000000018 < 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. Add Preprocessing

   3. Taylor expanded in x around inf 99.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\left(6.039053782637804 + \frac{1686.279566230464}{{x}^{2}}\right) - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   4. Step-by-step derivation

      1. associate--l+ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{6.039053782637804 + \left(\frac{1686.279566230464}{{x}^{2}} - 82.23527511657367 \cdot \frac{1}{x}\right)}}{x} - x\right) \]

      2. unpow2 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464}{\color{blue}{x \cdot x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      3. associate-/r* 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\color{blue}{\frac{\frac{1686.279566230464}{x}}{x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{\frac{\color{blue}{1686.279566230464 \cdot 1}}{x}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      5. associate-*r/ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x}}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      6. associate-*r/ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}\right)}{x} - x\right) \]

      7. metadata-eval 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \frac{\color{blue}{82.23527511657367}}{x}\right)}{x} - x\right) \]

      8. div-sub 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \color{blue}{\frac{1686.279566230464 \cdot \frac{1}{x} - 82.23527511657367}{x}}}{x} - x\right) \]

      9. sub-neg 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x} + \left(-82.23527511657367\right)}}{x}}{x} - x\right) \]

      10. associate-*r/ 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\color{blue}{\frac{1686.279566230464 \cdot 1}{x}} + \left(-82.23527511657367\right)}{x}}{x} - x\right) \]

      11. metadata-eval 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{\color{blue}{1686.279566230464}}{x} + \left(-82.23527511657367\right)}{x}}{x} - x\right) \]

      12. metadata-eval 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + \color{blue}{-82.23527511657367}}{x}}{x} - x\right) \]

   5. Simplified 99.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x}} - x\right) \]

   if -7.20000000000000018 < x < 6.20000000000000018

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + -1.7950336306565942 \cdot x\right) - 2.0191289437\right)\right)} - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot \left(x \cdot \left(1.900161040244073 + x \cdot -1.7950336306565942\right) - 2.0191289437\right)\right) - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -7.2) (not (<= x 6.2)))
   (*
    0.70711
    (-
     (/
      (+
       6.039053782637804
       (/ (+ (/ 1686.279566230464 x) -82.23527511657367) x))
      x)
     x))
   (+
    1.6316775383
    (*
     x
     (-
      (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
      2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -7.2) || !(x <= 6.2)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.2d0)) .or. (.not. (x <= 6.2d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 + (((1686.279566230464d0 / x) + (-82.23527511657367d0)) / x)) / x) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -7.2) || !(x <= 6.2)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -7.2) or not (x <= 6.2):
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -7.2) || !(x <= 6.2))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 + Float64(Float64(Float64(1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -7.2) || ~((x <= 6.2)))
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	else
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -7.2], N[Not[LessEqual[x, 6.2]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 + N[(N[(N[(1686.279566230464 / x), $MachinePrecision] + -82.23527511657367), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.20000000000000018 or 6.20000000000000018 < 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. Add Preprocessing

   3. Taylor expanded in x around inf 99.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{\left(6.039053782637804 + \frac{1686.279566230464}{{x}^{2}}\right) - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   4. Step-by-step derivation

      1. associate--l+ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{\color{blue}{6.039053782637804 + \left(\frac{1686.279566230464}{{x}^{2}} - 82.23527511657367 \cdot \frac{1}{x}\right)}}{x} - x\right) \]

      2. unpow2 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464}{\color{blue}{x \cdot x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      3. associate-/r* 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\color{blue}{\frac{\frac{1686.279566230464}{x}}{x}} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      4. metadata-eval 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{\frac{\color{blue}{1686.279566230464 \cdot 1}}{x}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      5. associate-*r/ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x}}}{x} - 82.23527511657367 \cdot \frac{1}{x}\right)}{x} - x\right) \]

      6. associate-*r/ 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}\right)}{x} - x\right) \]

      7. metadata-eval 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \left(\frac{1686.279566230464 \cdot \frac{1}{x}}{x} - \frac{\color{blue}{82.23527511657367}}{x}\right)}{x} - x\right) \]

      8. div-sub 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \color{blue}{\frac{1686.279566230464 \cdot \frac{1}{x} - 82.23527511657367}{x}}}{x} - x\right) \]

      9. sub-neg 99.0%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\color{blue}{1686.279566230464 \cdot \frac{1}{x} + \left(-82.23527511657367\right)}}{x}}{x} - x\right) \]

      10. associate-*r/ 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\color{blue}{\frac{1686.279566230464 \cdot 1}{x}} + \left(-82.23527511657367\right)}{x}}{x} - x\right) \]

      11. metadata-eval 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{\color{blue}{1686.279566230464}}{x} + \left(-82.23527511657367\right)}{x}}{x} - x\right) \]

      12. metadata-eval 99.0%

          \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + \color{blue}{-82.23527511657367}}{x}}{x} - x\right) \]

   5. Simplified 99.0%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x}} - x\right) \]

   if -7.20000000000000018 < x < 6.20000000000000018

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-define 99.9%

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

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \lor \neg \left(x \leq 6.2\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \lor \neg \left(x \leq 2.25\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.8) (not (<= x 2.25)))
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (+
    1.6316775383
    (*
     x
     (-
      (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
      2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.8) || !(x <= 2.25)) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.8d0)) .or. (.not. (x <= 2.25d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.8) || !(x <= 2.25)) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.8) or not (x <= 2.25):
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.8) || !(x <= 2.25))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.8) || ~((x <= 2.25)))
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	else
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.8], N[Not[LessEqual[x, 2.25]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999982 or 2.25 < 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x\right) \]

      2. metadata-eval 98.9%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x\right) \]

   5. Simplified 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x\right) \]

   if -4.79999999999999982 < x < 2.25

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-define 99.9%

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

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \lor \neg \left(x \leq 2.25\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -5.0) (not (<= x 1.15)))
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (* 0.70711 (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437))))))

C

double code(double x) {
	double tmp;
	if ((x <= -5.0) || !(x <= 1.15)) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} 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 <= (-5.0d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    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 <= -5.0) || !(x <= 1.15)) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else {
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -5.0) or not (x <= 1.15):
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	else:
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -5.0) || !(x <= 1.15))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	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 <= -5.0) || ~((x <= 1.15)))
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	else
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -5.0], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $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 < -5 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x\right) \]
   4. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x\right) \]

      2. metadata-eval 98.9%

         \[\leadsto 0.70711 \cdot \left(\frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x\right) \]

   5. Simplified 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x\right) \]

   if -5 < x < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.55)
     (* 0.70711 (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437))))
     (+ (/ 4.2702753202410175 x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.55) {
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 1.55d0) then
        tmp = 0.70711d0 * (2.30753d0 + (x * ((x * 1.900161040244073d0) - 3.0191289437d0)))
    else
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.55) {
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 1.55:
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)))
	else:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 1.55)
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) - 3.0191289437))));
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 1.55)
		tmp = 0.70711 * (2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437)));
	else
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55], N[(0.70711 * N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] - 3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.8%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -1.05000000000000004 < x < 1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 3.0191289437\right)\right)} \]

   if 1.55000000000000004 < 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   4. Taylor expanded in x around inf 98.7%

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

      1. sub-neg 98.7%

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

      2. metadata-eval 98.7%

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

      3. distribute-lft-in 98.7%

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

      4. associate-*r/ 98.7%

         \[\leadsto x \cdot \color{blue}{\frac{4.2702753202410175 \cdot 1}{{x}^{2}}} + x \cdot -0.70711 \]

      5. metadata-eval 98.7%

         \[\leadsto x \cdot \frac{\color{blue}{4.2702753202410175}}{{x}^{2}} + x \cdot -0.70711 \]

      6. associate-/l* 98.7%

         \[\leadsto \color{blue}{\frac{x \cdot 4.2702753202410175}{{x}^{2}}} + x \cdot -0.70711 \]

      7. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{4.2702753202410175 \cdot x}}{{x}^{2}} + x \cdot -0.70711 \]

      8. unpow2 98.7%

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

      9. times-frac 98.7%

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

      10. *-inverses 98.7%

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

      11. *-rgt-identity 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.55)
     (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))
     (+ (/ 4.2702753202410175 x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 1.55d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    else
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 1.55:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	else:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 1.55)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 1.55)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	else
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.8%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -1.05000000000000004 < x < 1.55000000000000004

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-define 99.9%

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

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(1.3436228731669864 \cdot x - 2.134856267379707\right)} \]

   if 1.55000000000000004 < 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   4. Taylor expanded in x around inf 98.7%

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

      1. sub-neg 98.7%

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

      2. metadata-eval 98.7%

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

      3. distribute-lft-in 98.7%

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

      4. associate-*r/ 98.7%

         \[\leadsto x \cdot \color{blue}{\frac{4.2702753202410175 \cdot 1}{{x}^{2}}} + x \cdot -0.70711 \]

      5. metadata-eval 98.7%

         \[\leadsto x \cdot \frac{\color{blue}{4.2702753202410175}}{{x}^{2}} + x \cdot -0.70711 \]

      6. associate-/l* 98.7%

         \[\leadsto \color{blue}{\frac{x \cdot 4.2702753202410175}{{x}^{2}}} + x \cdot -0.70711 \]

      7. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{4.2702753202410175 \cdot x}}{{x}^{2}} + x \cdot -0.70711 \]

      8. unpow2 98.7%

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

      9. times-frac 98.7%

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

      10. *-inverses 98.7%

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

      11. *-rgt-identity 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (* x -0.70711)
   (* 0.70711 (+ 2.30753 (* x -3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 0.70711 * (2.30753 + (x * -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 <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    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 = 0.70711 * (2.30753 + (x * -3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	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 = 0.70711 * (2.30753 + (x * -3.0191289437));
	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[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-lft-in 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-define 99.7%

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

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 0.70711 \cdot \left(2.30753 + \color{blue}{x \cdot -3.0191289437}\right) \]

   6. Simplified 99.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.8)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (* 0.70711 (+ 2.30753 (* x -3.0191289437)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 0.70711 * (2.30753 + (x * -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 <= 2.8d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.8):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.8))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.8)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -1.05000000000000004 < x < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 0.70711 \cdot \left(2.30753 + \color{blue}{x \cdot -3.0191289437}\right) \]

   6. Simplified 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 2.8)
     (* 0.70711 (+ 2.30753 (* x -3.0191289437)))
     (+ (/ 4.2702753202410175 x) (* x -0.70711)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.8) {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 2.8d0) then
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    else
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.8) {
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	} else {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 2.8:
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437))
	else:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 2.8)
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 2.8)
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	else
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8], N[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.8%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   if -1.05000000000000004 < x < 2.7999999999999998

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 99.2%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + -3.0191289437 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto 0.70711 \cdot \left(2.30753 + \color{blue}{x \cdot -3.0191289437}\right) \]

   6. Simplified 99.2%

      \[\leadsto 0.70711 \cdot \color{blue}{\left(2.30753 + x \cdot -3.0191289437\right)} \]

   if 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\frac{6.039053782637804}{x}} - x\right) \]

   4. Taylor expanded in x around inf 98.7%

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

      1. sub-neg 98.7%

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

      2. metadata-eval 98.7%

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

      3. distribute-lft-in 98.7%

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

      4. associate-*r/ 98.7%

         \[\leadsto x \cdot \color{blue}{\frac{4.2702753202410175 \cdot 1}{{x}^{2}}} + x \cdot -0.70711 \]

      5. metadata-eval 98.7%

         \[\leadsto x \cdot \frac{\color{blue}{4.2702753202410175}}{{x}^{2}} + x \cdot -0.70711 \]

      6. associate-/l* 98.7%

         \[\leadsto \color{blue}{\frac{x \cdot 4.2702753202410175}{{x}^{2}}} + x \cdot -0.70711 \]

      7. *-commutative 98.7%

         \[\leadsto \frac{\color{blue}{4.2702753202410175 \cdot x}}{{x}^{2}} + x \cdot -0.70711 \]

      8. unpow2 98.7%

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

      9. times-frac 98.7%

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

      10. *-inverses 98.7%

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

      11. *-rgt-identity 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(2.30753 + x \cdot -3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.8% 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 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-lft-in 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-define 99.7%

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

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. metadata-eval 99.9%

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

      9. associate-*r/ 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-lft-in 99.9%

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

      12. *-commutative 99.9%

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

      13. associate-*r* 99.9%

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

      14. *-commutative 99.9%

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

      15. fma-define 99.9%

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

      16. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.9%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot -2.134856267379707} \]
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 12: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		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 <= 1.15d0))) 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 <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	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
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		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 <= 1.15)))
		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, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

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. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-lft-in 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-define 99.7%

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

      16. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, \color{blue}{0.1913510371}, 0.70711 \cdot 2.30753\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      17. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, \color{blue}{1.6316775383}\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right) \]

      18. +-commutative 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right) \]

      19. fma-define 99.7%

          \[\leadsto \mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.70711, \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.5%

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

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   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. Add Preprocessing

   3. Taylor expanded in x around 0 99.7%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 97.8%

      \[\leadsto 0.70711 \cdot \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

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

Alternative 13: 50.3% 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 49.9%

   \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x\right) \]

4. Taylor expanded in x around 0 48.7%

   \[\leadsto 0.70711 \cdot \color{blue}{2.30753} \]

5. Final simplification 48.7%

   \[\leadsto 1.6316775383 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(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)))