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

Percentage Accurate: 99.9% → 99.9%

Time: 13.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(x * -0.70711 + N[(N[(x * 0.1913510371 + 1.6316775383), $MachinePrecision] / N[(x * N[(x * 0.04481 + 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-rgt-in 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. associate-*l/ 99.9%

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

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. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.8) (not (<= x 1.15)))
   (* 0.70711 (- (- (/ 6.039053782637804 x) (/ 82.23527511657367 (* x x))) x))
   (+ 1.6316775383 (* x (+ (* x 1.4167383436582095) -2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.8) || !(x <= 1.15)) {
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.8d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 / x) - (82.23527511657367d0 / (x * x))) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * 1.4167383436582095d0) + (-2.134856267379707d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.8) || !(x <= 1.15)) {
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.8) or not (x <= 1.15):
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.8) || !(x <= 1.15))
		tmp = Float64(0.70711 * Float64(Float64(Float64(6.039053782637804 / x) - Float64(82.23527511657367 / Float64(x * x))) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.4167383436582095) + -2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.8) || ~((x <= 1.15)))
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * x))) - x);
	else
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.8], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(0.70711 * N[(N[(N[(6.039053782637804 / x), $MachinePrecision] - N[(82.23527511657367 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * 1.4167383436582095), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999982 or 1.1499999999999999 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

      3. unpow2 99.2%

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

      4. associate-*r/ 99.2%

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

      5. metadata-eval 99.2%

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

   4. Simplified 99.2%

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

   if -4.79999999999999982 < 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. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

      4. associate-*l* 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-lft-out 98.6%

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

   7. Simplified 98.6%

      \[\leadsto \color{blue}{1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.4167383436582095 + -2.134856267379707\right)\\ \end{array} \]

Alternative 4: 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 99.8%

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

2. Taylor expanded in x around 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 5: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (if (<= x 2.8)
     (* 0.70711 (- (+ 2.30753 (* x -2.0191289437)) x))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 2.8) {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else if (x <= 2.8d0) then
        tmp = 0.70711d0 * ((2.30753d0 + (x * (-2.0191289437d0))) - x)
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 2.8) {
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	elif x <= 2.8:
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x)
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	elseif (x <= 2.8)
		tmp = Float64(0.70711 * Float64(Float64(2.30753 + Float64(x * -2.0191289437)) - x));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	elseif (x <= 2.8)
		tmp = 0.70711 * ((2.30753 + (x * -2.0191289437)) - x);
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8], N[(0.70711 * N[(N[(2.30753 + N[(x * -2.0191289437), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. 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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

   if -1.05000000000000004 < x < 2.7999999999999998

   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 98.5%

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

   if 2.7999999999999998 < 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;0.70711 \cdot \left(\left(2.30753 + x \cdot -2.0191289437\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (if (<= x 1.55)
     (+ 1.6316775383 (* x (+ (* x 1.4167383436582095) -2.134856267379707)))
     (* 0.70711 (- (/ 6.039053782637804 x) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else if (x <= 1.55d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.4167383436582095d0) + (-2.134856267379707d0)))
    else
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 1.55) {
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	} else {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	elif x <= 1.55:
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707))
	else:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	elseif (x <= 1.55)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.4167383436582095) + -2.134856267379707)));
	else
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	elseif (x <= 1.55)
		tmp = 1.6316775383 + (x * ((x * 1.4167383436582095) + -2.134856267379707));
	else
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55], N[(1.6316775383 + N[(x * N[(N[(x * 1.4167383436582095), $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 < -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. 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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

   if -1.05000000000000004 < x < 1.55000000000000004

   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 98.6%

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

      1. *-commutative 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. unpow2 98.6%

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

      4. associate-*l* 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-lft-out 98.6%

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

   7. Simplified 98.6%

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

   if 1.55000000000000004 < 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 98.9%

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

Alternative 7: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(0.70711 * Float64(2.30753 + Float64(x * -3.0191289437)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = x * -0.70711;
	else
		tmp = 0.70711 * (2.30753 + (x * -3.0191289437));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   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 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x\right) \]

      2. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      3. fma-def 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      4. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right)\right) - x\right) \]

      5. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)} + 1}\right)\right) - x\right) \]

      6. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)} + 1}\right)\right) - x\right) \]

      7. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right)\right) - x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.7%

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

Alternative 8: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 2.8d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 0.70711d0 * (2.30753d0 + (x * (-3.0191289437d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.7999999999999998 < x

   1. Initial program 99.8%

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

   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 < 2.7999999999999998

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x\right) \]

      2. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      3. fma-def 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      4. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right)\right) - x\right) \]

      5. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)} + 1}\right)\right) - x\right) \]

      6. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)} + 1}\right)\right) - x\right) \]

      7. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right)\right) - x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.8%

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

Alternative 9: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8], N[(0.70711 * N[(2.30753 + N[(x * -3.0191289437), $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 < -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. 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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.1%

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

   5. Taylor expanded in x around 0 99.1%

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

   if -1.05000000000000004 < x < 2.7999999999999998

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

         \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right)} - x\right) \]

      2. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      3. fma-def 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\right)\right) - x\right) \]

      4. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}}\right)\right) - x\right) \]

      5. +-commutative 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)} + 1}\right)\right) - x\right) \]

      6. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)} + 1}\right)\right) - x\right) \]

      7. fma-udef 99.9%

         \[\leadsto 0.70711 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}\right)\right) - x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto 0.70711 \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)\right)} - x\right) \]

   4. Taylor expanded in x around 0 98.5%

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

      1. *-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 2.7999999999999998 < 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 98.8%

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

Alternative 10: 98.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* x -0.70711)
   (if (<= x 1.15) (+ 1.6316775383 (* x -2.134856267379707)) (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. 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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   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 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 99.9%

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

   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 98.5%

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

      1. *-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 11: 98.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.4) (* x -0.70711) (if (<= x 1.15) 1.6316775383 (* x -0.70711))))

C

double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.4d0)) then
        tmp = x * (-0.70711d0)
    else if (x <= 1.15d0) then
        tmp = 1.6316775383d0
    else
        tmp = x * (-0.70711d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.4) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = 1.6316775383;
	} else {
		tmp = x * -0.70711;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.4:
		tmp = x * -0.70711
	elif x <= 1.15:
		tmp = 1.6316775383
	else:
		tmp = x * -0.70711
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.4)
		tmp = Float64(x * -0.70711);
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = Float64(x * -0.70711);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.4)
		tmp = x * -0.70711;
	elseif (x <= 1.15)
		tmp = 1.6316775383;
	else
		tmp = x * -0.70711;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.4], N[(x * -0.70711), $MachinePrecision], If[LessEqual[x, 1.15], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999991 or 1.1499999999999999 < x

   1. Initial program 99.8%

      \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
   2. 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-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   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 -3.39999999999999991 < 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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 99.9%

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

   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. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. flip-+ 100.0%

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

      3. pow2 100.0%

         \[\leadsto \frac{\left(x \cdot -0.70711\right) \cdot \left(x \cdot -0.70711\right) - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\left(x \cdot -0.70711\right) \cdot \left(x \cdot -0.70711\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]
   6. Step-by-step derivation

      1. swap-sqr 100.0%

         \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(-0.70711 \cdot -0.70711\right)} - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

      2. associate-*l* 100.0%

         \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(-0.70711 \cdot -0.70711\right)\right)} - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{x \cdot \left(x \cdot \color{blue}{0.5000045521}\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot 0.5000045521\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

   8. Taylor expanded in x around 0 97.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 12: 50.5% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.8%

   \[0.70711 \cdot \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
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-rgt-in 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. associate-*l/ 99.9%

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

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. Step-by-step derivation

   1. fma-udef 99.9%

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

   2. flip-+ 76.9%

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

   3. pow2 76.9%

      \[\leadsto \frac{\left(x \cdot -0.70711\right) \cdot \left(x \cdot -0.70711\right) - \color{blue}{{\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

5. Applied egg-rr 76.9%

   \[\leadsto \color{blue}{\frac{\left(x \cdot -0.70711\right) \cdot \left(x \cdot -0.70711\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]
6. Step-by-step derivation

   1. swap-sqr 76.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x\right) \cdot \left(-0.70711 \cdot -0.70711\right)} - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

   2. associate-*l* 76.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(-0.70711 \cdot -0.70711\right)\right)} - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

   3. metadata-eval 76.7%

      \[\leadsto \frac{x \cdot \left(x \cdot \color{blue}{0.5000045521}\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]

7. Simplified 76.7%

   \[\leadsto \color{blue}{\frac{x \cdot \left(x \cdot 0.5000045521\right) - {\left(\frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}\right)}^{2}}{x \cdot -0.70711 - \frac{\mathsf{fma}\left(x, 0.1913510371, 1.6316775383\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}}} \]

8. Taylor expanded in x around 0 47.0%

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

9. Final simplification 47.0%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023279 
(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)))