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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot \left(-2.0191289437 + x \cdot 2.003561459544073\right)\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.55)
     (*
      0.70711
      (- (+ 2.30753 (* x (+ -2.0191289437 (* x 2.003561459544073)))) x))
     (+ (/ 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 * (-2.0191289437 + (x * 2.003561459544073)))) - x);
	} 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 * ((-2.0191289437d0) + (x * 2.003561459544073d0)))) - x)
    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 * (-2.0191289437 + (x * 2.003561459544073)))) - x);
	} 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 * (-2.0191289437 + (x * 2.003561459544073)))) - x)
	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(Float64(2.30753 + Float64(x * Float64(-2.0191289437 + Float64(x * 2.003561459544073)))) - x));
	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 * (-2.0191289437 + (x * 2.003561459544073)))) - x);
	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[(N[(2.30753 + N[(x * N[(-2.0191289437 + N[(x * 2.003561459544073), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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 99.8%

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

   2. Taylor expanded in x around inf 99.1%

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

   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. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + \left(-2.0191289437 \cdot x + 2.003561459544073 \cdot {x}^{2}\right)\right)} - x\right) \]
   6. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 0.70711 \cdot \left(\left(2.30753 + \left(\color{blue}{x \cdot -2.0191289437} + 2.003561459544073 \cdot {x}^{2}\right)\right) - x\right) \]

      2. *-commutative 98.9%

         \[\leadsto 0.70711 \cdot \left(\left(2.30753 + \left(x \cdot -2.0191289437 + \color{blue}{{x}^{2} \cdot 2.003561459544073}\right)\right) - x\right) \]

      3. unpow2 98.9%

         \[\leadsto 0.70711 \cdot \left(\left(2.30753 + \left(x \cdot -2.0191289437 + \color{blue}{\left(x \cdot x\right)} \cdot 2.003561459544073\right)\right) - x\right) \]

      4. associate-*l* 98.9%

         \[\leadsto 0.70711 \cdot \left(\left(2.30753 + \left(x \cdot -2.0191289437 + \color{blue}{x \cdot \left(x \cdot 2.003561459544073\right)}\right)\right) - x\right) \]

      5. distribute-lft-out 98.9%

         \[\leadsto 0.70711 \cdot \left(\left(2.30753 + \color{blue}{x \cdot \left(-2.0191289437 + x \cdot 2.003561459544073\right)}\right) - x\right) \]

   7. Simplified 98.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\left(2.30753 + x \cdot \left(-2.0191289437 + x \cdot 2.003561459544073\right)\right)} - x\right) \]

   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. 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. neg-mul-1 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      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. distribute-lft-in 99.7%

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

      11. +-commutative 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%

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

      13. associate-*l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. fma-def 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

      19. +-commutative 99.7%

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

      20. fma-def 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. Taylor expanded in x around inf 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot \left(-2.0191289437 + x \cdot 2.003561459544073\right)\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

4. Simplified 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.55)
     (+ 1.6316775383 (* x (+ (* x 1.4167383436582095) -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 * 1.4167383436582095) + -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 * 1.4167383436582095d0) + (-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 * 1.4167383436582095) + -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 * 1.4167383436582095) + -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 * 1.4167383436582095) + -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 * 1.4167383436582095) + -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 * 1.4167383436582095), $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 99.8%

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

   2. Taylor expanded in x around inf 99.1%

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

   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. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{1.6316775383 + \left(-2.134856267379707 \cdot x + 1.4167383436582095 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 98.9%

         \[\leadsto 1.6316775383 + \color{blue}{\left(1.4167383436582095 \cdot {x}^{2} + -2.134856267379707 \cdot x\right)} \]

      2. *-commutative 98.9%

         \[\leadsto 1.6316775383 + \left(\color{blue}{{x}^{2} \cdot 1.4167383436582095} + -2.134856267379707 \cdot x\right) \]

      3. unpow2 98.9%

         \[\leadsto 1.6316775383 + \left(\color{blue}{\left(x \cdot x\right)} \cdot 1.4167383436582095 + -2.134856267379707 \cdot x\right) \]

      4. associate-*l* 98.9%

         \[\leadsto 1.6316775383 + \left(\color{blue}{x \cdot \left(x \cdot 1.4167383436582095\right)} + -2.134856267379707 \cdot x\right) \]

      5. *-commutative 98.9%

         \[\leadsto 1.6316775383 + \left(x \cdot \left(x \cdot 1.4167383436582095\right) + \color{blue}{x \cdot -2.134856267379707}\right) \]

      6. distribute-lft-out 98.9%

         \[\leadsto 1.6316775383 + \color{blue}{x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)} \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)} \]

   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. 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. neg-mul-1 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      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. distribute-lft-in 99.7%

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

      11. +-commutative 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%

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

      13. associate-*l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. fma-def 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

      19. +-commutative 99.7%

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

      20. fma-def 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. Taylor expanded in x around inf 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 0.74)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x -2.134856267379707))))

C

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

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 0.74):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 0.74))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	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 <= 0.74)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	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, 0.74]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.73999999999999999 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 99.3%

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

   if -1.05000000000000004 < x < 0.73999999999999999

   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. neg-mul-1 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. fma-def 99.9%

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

      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. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*l* 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. fma-def 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

      20. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 99.1%

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

Alternative 6: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 0.74)
     (+ 1.6316775383 (* x -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 <= 0.74) {
		tmp = 1.6316775383 + (x * -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 <= 0.74d0) then
        tmp = 1.6316775383d0 + (x * (-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 <= 0.74) {
		tmp = 1.6316775383 + (x * -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 <= 0.74:
		tmp = 1.6316775383 + (x * -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 <= 0.74)
		tmp = Float64(1.6316775383 + Float64(x * -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 <= 0.74)
		tmp = 1.6316775383 + (x * -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, 0.74], N[(1.6316775383 + N[(x * -2.134856267379707), $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 99.8%

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

   2. Taylor expanded in x around inf 99.1%

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

   if -1.05000000000000004 < x < 0.73999999999999999

   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. neg-mul-1 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. fma-def 99.9%

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

      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. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*l* 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. fma-def 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

      20. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

   if 0.73999999999999999 < 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. neg-mul-1 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      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. distribute-lft-in 99.7%

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

      11. +-commutative 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%

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

      13. associate-*l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. fma-def 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

      19. +-commutative 99.7%

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

      20. fma-def 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. Taylor expanded in x around inf 99.5%

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

   5. Taylor expanded in x around 0 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 0.74:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \end{array} \]

Alternative 7: 98.8% accurate, 2.1× speedup

Math

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

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. neg-mul-1 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      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. distribute-lft-in 99.7%

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

      11. +-commutative 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%

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

      13. associate-*l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. fma-def 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

      19. +-commutative 99.7%

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

      20. fma-def 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. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   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. neg-mul-1 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. fma-def 99.9%

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

      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. distribute-lft-in 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*l* 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

      16. fma-def 100.0%

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

      17. metadata-eval 100.0%

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

      18. metadata-eval 100.0%

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

      19. +-commutative 100.0%

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

      20. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 98.8%

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

      1. *-commutative 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 98.8%

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

Alternative 8: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.2))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(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 <= (-1.05d0)) .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 <= -1.05) || !(x <= 1.2)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.2):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(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 <= -1.05) || ~((x <= 1.2)))
		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, 1.2]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 99.7%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. 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. neg-mul-1 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. fma-def 99.7%

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

      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. distribute-lft-in 99.7%

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

      11. +-commutative 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%

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

      13. associate-*l* 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

      16. fma-def 99.7%

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

      17. metadata-eval 99.7%

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

      18. metadata-eval 99.7%

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

      19. +-commutative 99.7%

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

      20. fma-def 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. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around 0 97.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]

Alternative 9: 50.7% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 98.3%

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

   1. *-commutative 98.3%

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

4. Simplified 98.3%

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

5. Taylor expanded in x around 0 49.3%

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

6. Final simplification 49.3%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023310 
(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)))