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

Percentage Accurate: 99.9% → 99.9%

Time: 10.7s

Alternatives: 9

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, 0.2× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{x \cdot 0.27061 + 2.30753}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot {x}^{2}} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (/
    (+ (* x 0.27061) 2.30753)
    (+ (+ 1.0 (* x 0.99229)) (* 0.04481 (pow x 2.0))))
   x)))

C

double code(double x) {
	return 0.70711 * ((((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * pow(x, 2.0)))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * ((((x * 0.27061d0) + 2.30753d0) / ((1.0d0 + (x * 0.99229d0)) + (0.04481d0 * (x ** 2.0d0)))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * ((((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * Math.pow(x, 2.0)))) - x);
}

Python

def code(x):
	return 0.70711 * ((((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * math.pow(x, 2.0)))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(1.0 + Float64(x * 0.99229)) + Float64(0.04481 * (x ^ 2.0)))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * ((((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * (x ^ 2.0)))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision] + N[(0.04481 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-define 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

   2. fma-undefine 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-rgt-in 99.9%

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

   6. associate-+r+ 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. associate-*l* 99.9%

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

   10. pow2 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.0)
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (if (<= x 1.1)
     (+
      1.6316775383
      (*
       x
       (-
        (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
        2.134856267379707)))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 1.1) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    else if (x <= 1.1d0) then
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 1.1) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.0:
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	elif x <= 1.1:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	elseif (x <= 1.1)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	elseif (x <= 1.1)
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.0], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   if -5 < x < 1.1000000000000001

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + -1.2692862305735844 \cdot x\right) - 2.134856267379707\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.0)
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (if (<= x 1.2)
     (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 1.2) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    else if (x <= 1.2d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 1.2) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.0:
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	elif x <= 1.2:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	elseif (x <= 1.2)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	elseif (x <= 1.2)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.0], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   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 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   5. Simplified 99.4%

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

   if -5 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.1%

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

   if 1.19999999999999996 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (* x -0.70711) (/ 4.2702753202410175 x))
   (if (<= x 1.2)
     (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))
     (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.2) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (x * (-0.70711d0)) + (4.2702753202410175d0 / x)
    else if (x <= 1.2d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	} else if (x <= 1.2) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (x * -0.70711) + (4.2702753202410175 / x)
	elif x <= 1.2:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(x * -0.70711) + Float64(4.2702753202410175 / x));
	elseif (x <= 1.2)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	elseif (x <= 1.2)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(x * -0.70711), $MachinePrecision] + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $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. Step-by-step derivation

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-define 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

      2. fma-undefine 99.7%

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

      3. +-commutative 99.7%

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

      4. +-commutative 99.7%

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

      5. distribute-rgt-in 99.7%

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

      6. associate-+r+ 99.7%

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

      7. *-commutative 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-*l* 99.7%

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

      10. pow2 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{x \cdot \left(4.2702753202410175 \cdot \frac{1}{{x}^{2}} - 0.70711\right)} \]
   8. 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. unpow2 98.7%

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

      8. times-frac 98.7%

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

      9. *-inverses 98.7%

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

      10. *-lft-identity 98.7%

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

   9. Simplified 98.7%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 98.1%

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

   if 1.19999999999999996 < x

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 98.0%

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

   1. *-commutative 98.0%

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

5. Simplified 98.0%

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

6. Final simplification 98.0%

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

Alternative 7: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.5) (not (<= x 1.2))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.5d0)) .or. (.not. (x <= 1.2d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.5) or not (x <= 1.2):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.5) || !(x <= 1.2))
		tmp = Float64(x * -0.70711);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.5) || ~((x <= 1.2)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.5], N[Not[LessEqual[x, 1.2]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.19999999999999996 < x

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-define 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. fma-undefine 99.8%

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

      2. fma-undefine 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. associate-+r+ 99.8%

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

      7. *-commutative 99.8%

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

      8. *-commutative 99.8%

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

      9. associate-*l* 99.8%

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

      10. pow2 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   9. Simplified 98.9%

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

   if -3.5 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. fma-undefine 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-rgt-in 99.9%

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

      6. associate-+r+ 99.9%

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

      7. *-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-*l* 99.9%

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

      10. pow2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 95.9%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 0.70711 (- 2.30753 x)))

C

double code(double x) {
	return 0.70711 * (2.30753 - x);
}

Fortran

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

Java

public static double code(double x) {
	return 0.70711 * (2.30753 - x);
}

Python

def code(x):
	return 0.70711 * (2.30753 - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(2.30753 - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * (2.30753 - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(2.30753 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 96.9%

   \[\leadsto 0.70711 \cdot \left(\color{blue}{2.30753} - x\right) \]
4. Add Preprocessing

Alternative 9: 50.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 1.6316775383 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.6316775383d0
end function

Java

public static double code(double x) {
	return 1.6316775383;
}

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

function tmp = code(x)
	tmp = 1.6316775383;
end

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

   3. +-commutative 99.9%

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

   4. fma-define 99.9%

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

   5. +-commutative 99.9%

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

   6. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

   2. fma-undefine 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. distribute-rgt-in 99.9%

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

   6. associate-+r+ 99.9%

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

   7. *-commutative 99.9%

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

   8. *-commutative 99.9%

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

   9. associate-*l* 99.9%

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

   10. pow2 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 50.3%

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

Reproduce

herbie shell --seed 2024085 
(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)))