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

Percentage Accurate: 99.8% → 99.9%

Time: 16.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.70711, \frac{x \cdot 0.1913510371 + 1.6316775383}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  x
  -0.70711
  (/
   (+ (* x 0.1913510371) 1.6316775383)
   (fma x (+ (* x 0.04481) 0.99229) 1.0))))

C

double code(double x) {
	return fma(x, -0.70711, (((x * 0.1913510371) + 1.6316775383) / fma(x, ((x * 0.04481) + 0.99229), 1.0)));
}

Julia

function code(x)
	return fma(x, -0.70711, Float64(Float64(Float64(x * 0.1913510371) + 1.6316775383) / fma(x, Float64(Float64(x * 0.04481) + 0.99229), 1.0)))
end

Wolfram

code[x_] := N[(x * -0.70711 + N[(N[(N[(x * 0.1913510371), $MachinePrecision] + 1.6316775383), $MachinePrecision] / N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. distribute-rgt-neg-out 99.8%

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

   5. *-commutative 99.8%

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

   6. distribute-rgt-neg-in 99.8%

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

   7. fma-define 99.8%

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

   8. metadata-eval 99.8%

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

   9. associate-*r/ 99.8%

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

   10. +-commutative 99.8%

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

   11. distribute-lft-in 99.9%

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

   12. *-commutative 99.9%

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

   13. associate-*r* 99.9%

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

   14. *-commutative 99.9%

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

   15. fma-define 99.9%

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

   16. metadata-eval 99.9%

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

   17. metadata-eval 99.9%

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

   18. +-commutative 99.9%

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

   19. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x}\\ \mathbf{if}\;x \leq -7.1:\\ \;\;\;\;0.70711 \cdot \left(t\_0 - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot t\_0 + x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (/
          (+
           6.039053782637804
           (/ (+ (/ 1686.279566230464 x) -82.23527511657367) x))
          x)))
   (if (<= x -7.1)
     (* 0.70711 (- t_0 x))
     (if (<= x 4.8e-8)
       (+
        1.6316775383
        (*
         x
         (-
          (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
          2.134856267379707)))
       (+ (* 0.70711 t_0) (* x -0.70711))))))

C

double code(double x) {
	double t_0 = (6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x;
	double tmp;
	if (x <= -7.1) {
		tmp = 0.70711 * (t_0 - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = (0.70711 * t_0) + (x * -0.70711);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (6.039053782637804d0 + (((1686.279566230464d0 / x) + (-82.23527511657367d0)) / x)) / x
    if (x <= (-7.1d0)) then
        tmp = 0.70711d0 * (t_0 - x)
    else if (x <= 4.8d-8) then
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    else
        tmp = (0.70711d0 * t_0) + (x * (-0.70711d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x;
	double tmp;
	if (x <= -7.1) {
		tmp = 0.70711 * (t_0 - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = (0.70711 * t_0) + (x * -0.70711);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x
	tmp = 0
	if x <= -7.1:
		tmp = 0.70711 * (t_0 - x)
	elif x <= 4.8e-8:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	else:
		tmp = (0.70711 * t_0) + (x * -0.70711)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(6.039053782637804 + Float64(Float64(Float64(1686.279566230464 / x) + -82.23527511657367) / x)) / x)
	tmp = 0.0
	if (x <= -7.1)
		tmp = Float64(0.70711 * Float64(t_0 - x));
	elseif (x <= 4.8e-8)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	else
		tmp = Float64(Float64(0.70711 * t_0) + Float64(x * -0.70711));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x;
	tmp = 0.0;
	if (x <= -7.1)
		tmp = 0.70711 * (t_0 - x);
	elseif (x <= 4.8e-8)
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	else
		tmp = (0.70711 * t_0) + (x * -0.70711);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(6.039053782637804 + N[(N[(N[(1686.279566230464 / x), $MachinePrecision] + -82.23527511657367), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -7.1], N[(0.70711 * N[(t$95$0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-8], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.70711 * t$95$0), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0999999999999996

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.3%

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

      1. associate--l+ 99.3%

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

      2. unpow2 99.3%

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

      3. associate-/r* 99.3%

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

      4. metadata-eval 99.3%

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

      5. associate-*r/ 99.3%

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

      6. associate-*r/ 99.3%

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

      7. metadata-eval 99.3%

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

      8. div-sub 99.3%

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

      9. sub-neg 99.3%

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

      10. associate-*r/ 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

   5. Simplified 99.3%

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

   if -7.0999999999999996 < x < 4.79999999999999997e-8

   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 99.7%

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

   if 4.79999999999999997e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 95.5%

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

      1. associate--l+ 95.5%

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

      2. unpow2 95.5%

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

      3. associate-/r* 95.5%

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

      4. metadata-eval 95.5%

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

      5. associate-*r/ 95.5%

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

      6. associate-*r/ 95.5%

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

      7. metadata-eval 95.5%

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

      8. div-sub 95.5%

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

      9. sub-neg 95.5%

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

      10. associate-*r/ 95.5%

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

      11. metadata-eval 95.5%

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

      12. metadata-eval 95.5%

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

   5. Simplified 95.5%

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

      1. sub-neg 95.5%

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

      2. distribute-rgt-in 95.5%

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

      3. +-commutative 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in x around 0 95.5%

      \[\leadsto \frac{\frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x} + 6.039053782637804}{x} \cdot 0.70711 + \color{blue}{-0.70711 \cdot x} \]
   9. Step-by-step derivation

      1. *-commutative 95.5%

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

   10. Simplified 95.5%

       \[\leadsto \frac{\frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x} + 6.039053782637804}{x} \cdot 0.70711 + \color{blue}{x \cdot -0.70711} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.1:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} + x \cdot -0.70711\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.1 \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -7.1) (not (<= x 4.8e-8)))
   (*
    0.70711
    (-
     (/
      (+
       6.039053782637804
       (/ (+ (/ 1686.279566230464 x) -82.23527511657367) x))
      x)
     x))
   (+
    1.6316775383
    (*
     x
     (-
      (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
      2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -7.1) || !(x <= 4.8e-8)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-7.1d0)) .or. (.not. (x <= 4.8d-8))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 + (((1686.279566230464d0 / x) + (-82.23527511657367d0)) / x)) / x) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -7.1) || !(x <= 4.8e-8)) {
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -7.1) or not (x <= 4.8e-8):
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -7.1) || !(x <= 4.8e-8))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 + Float64(Float64(Float64(1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -7.1) || ~((x <= 4.8e-8)))
		tmp = 0.70711 * (((6.039053782637804 + (((1686.279566230464 / x) + -82.23527511657367) / x)) / x) - x);
	else
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -7.1], N[Not[LessEqual[x, 4.8e-8]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 + N[(N[(N[(1686.279566230464 / x), $MachinePrecision] + -82.23527511657367), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0999999999999996 or 4.79999999999999997e-8 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 97.2%

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

      1. associate--l+ 97.2%

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

      2. unpow2 97.2%

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

      3. associate-/r* 97.2%

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

      4. metadata-eval 97.2%

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

      5. associate-*r/ 97.2%

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

      6. associate-*r/ 97.2%

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

      7. metadata-eval 97.2%

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

      8. div-sub 97.2%

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

      9. sub-neg 97.2%

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

      10. associate-*r/ 97.2%

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

      11. metadata-eval 97.2%

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

      12. metadata-eval 97.2%

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

   5. Simplified 97.2%

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

   if -7.0999999999999996 < x < 4.79999999999999997e-8

   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 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.1 \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 + \frac{\frac{1686.279566230464}{x} + -82.23527511657367}{x}}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.8)
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (if (<= x 4.8e-8)
     (+
      1.6316775383
      (*
       x
       (-
        (* x (+ 1.3436228731669864 (* x -1.2692862305735844)))
        2.134856267379707)))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.8) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.8d0)) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    else if (x <= 4.8d-8) then
        tmp = 1.6316775383d0 + (x * ((x * (1.3436228731669864d0 + (x * (-1.2692862305735844d0)))) - 2.134856267379707d0))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.8) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.8:
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	elif x <= 4.8e-8:
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707))
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.8)
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x));
	elseif (x <= 4.8e-8)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * Float64(1.3436228731669864 + Float64(x * -1.2692862305735844))) - 2.134856267379707)));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.8)
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	elseif (x <= 4.8e-8)
		tmp = 1.6316775383 + (x * ((x * (1.3436228731669864 + (x * -1.2692862305735844))) - 2.134856267379707));
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.8], N[(0.70711 * N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e-8], N[(1.6316775383 + N[(x * N[(N[(x * N[(1.3436228731669864 + N[(x * -1.2692862305735844), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999982

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   if -4.79999999999999982 < x < 4.79999999999999997e-8

   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 99.7%

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

   if 4.79999999999999997e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 95.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot \left(1.3436228731669864 + x \cdot -1.2692862305735844\right) - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 4.8e-8)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 4.8e-8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 4.8d-8))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 4.8e-8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 4.8e-8):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 4.8e-8))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 4.8e-8)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 4.8e-8]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 4.79999999999999997e-8 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 97.0%

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

   if -1.05000000000000004 < x < 4.79999999999999997e-8

   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 99.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.0)
   (* 0.70711 (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x))
   (if (<= x 4.8e-8)
     (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = 0.70711d0 * (((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x)
    else if (x <= 4.8d-8) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.0:
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x)
	elif x <= 4.8e-8:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	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 <= 4.8e-8)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = 0.70711 * (((6.039053782637804 - (82.23527511657367 / x)) / x) - x);
	elseif (x <= 4.8e-8)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	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, 4.8e-8], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   5. Simplified 99.1%

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

   if -5 < x < 4.79999999999999997e-8

   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 99.5%

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

   if 4.79999999999999997e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 95.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 0.70711 * (((2.30753 + (x * 0.27061)) / (1.0 + (x * ((x * 0.04481) + 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 * ((x * 0.04481d0) + 0.99229d0)))) - x)
end function

Java

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

Python

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 8: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+37}:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1e+37)
   (* x -0.70711)
   (if (<= x 4.8e-8)
     (+ 1.6316775383 (* x -2.134856267379707))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1e+37) {
		tmp = x * -0.70711;
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d+37)) then
        tmp = x * (-0.70711d0)
    else if (x <= 4.8d-8) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1e+37) {
		tmp = x * -0.70711;
	} else if (x <= 4.8e-8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1e+37:
		tmp = x * -0.70711
	elif x <= 4.8e-8:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1e+37)
		tmp = Float64(x * -0.70711);
	elseif (x <= 4.8e-8)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1e+37)
		tmp = x * -0.70711;
	elseif (x <= 4.8e-8)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1e+37], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 4.8e-8], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e37

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   if -1.1e37 < x < 4.79999999999999997e-8

   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 96.2%

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

      1. *-commutative 96.2%

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

   5. Simplified 96.2%

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

   if 4.79999999999999997e-8 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 95.5%

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

Alternative 9: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+37} \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1e+37) (not (<= x 1.15)))
   (* x -0.70711)
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1e+37) || !(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 <= (-1.1d+37)) .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 <= -1.1e+37) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1e+37) 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 <= -1.1e+37) || !(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 <= -1.1e+37) || ~((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, -1.1e+37], 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 < -1.1e37 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   if -1.1e37 < x < 1.1499999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.0%

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

      1. *-commutative 96.0%

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

   5. Simplified 96.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+37} \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+37} \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1e+37) (not (<= x 4.8e-8))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1e+37) || !(x <= 4.8e-8)) {
		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 <= (-1.1d+37)) .or. (.not. (x <= 4.8d-8))) 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 <= -1.1e+37) || !(x <= 4.8e-8)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1e+37) or not (x <= 4.8e-8):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1e+37) || !(x <= 4.8e-8))
		tmp = Float64(x * -0.70711);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1e+37) || ~((x <= 4.8e-8)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1e+37], N[Not[LessEqual[x, 4.8e-8]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e37 or 4.79999999999999997e-8 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 97.0%

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

      1. *-commutative 97.0%

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

   5. Simplified 97.0%

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

   if -1.1e37 < x < 4.79999999999999997e-8

   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 94.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+37} \lor \neg \left(x \leq 4.8 \cdot 10^{-8}\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 52.7%

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

Reproduce

herbie shell --seed 2024107 
(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)))