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

Percentage Accurate: 99.8% → 99.9%

Time: 6.8s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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-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. Step-by-step derivation

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

   1. fma-udef 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 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. Final simplification 99.8%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.2)))
   (* 0.70711 (- (- (/ 6.039053782637804 x) (/ 82.23527511657367 (* x x))) x))
   (+ 1.6316775383 (* x (+ (* x 1.3436228731669864) -2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.2)) {
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * 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 <= 1.2d0))) then
        tmp = 0.70711d0 * (((6.039053782637804d0 / x) - (82.23527511657367d0 / (x * 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 <= 1.2)) {
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * 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 <= 1.2):
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * 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 <= 1.2))
		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 * 1.3436228731669864) + -2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.2)))
		tmp = 0.70711 * (((6.039053782637804 / x) - (82.23527511657367 / (x * 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, 1.2]], $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 * 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

      3. unpow2 98.6%

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

      4. associate-*r/ 98.6%

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

      5. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   1. Initial program 99.9%

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

      1. 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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{1.6316775383 + \left(-2.134856267379707 \cdot x + 1.3436228731669864 \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 1.6316775383 + \color{blue}{\left(1.3436228731669864 \cdot {x}^{2} + -2.134856267379707 \cdot x\right)} \]

      2. unpow2 99.0%

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

      3. associate-*r* 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. *-commutative 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.2\right):\\ \;\;\;\;0.70711 \cdot \left(\left(\frac{6.039053782637804}{x} - \frac{82.23527511657367}{x \cdot x}\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 1.56)
     (+ 1.6316775383 (* x (+ (* x 1.3436228731669864) -2.134856267379707)))
     (+ (* x -0.70711) (/ 4.2702753202410175 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.56) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 1.56d0) then
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) + (-2.134856267379707d0)))
    else
        tmp = (x * (-0.70711d0)) + (4.2702753202410175d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 1.56) {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	} else {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 1.56:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707))
	else:
		tmp = (x * -0.70711) + (4.2702753202410175 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 1.56)
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) + -2.134856267379707)));
	else
		tmp = Float64(Float64(x * -0.70711) + Float64(4.2702753202410175 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 1.56)
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) + -2.134856267379707));
	else
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.56], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] + -2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.70711), $MachinePrecision] + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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. Taylor expanded in x around inf 97.4%

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

   if -1.05000000000000004 < x < 1.5600000000000001

   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%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{1.6316775383 + \left(-2.134856267379707 \cdot x + 1.3436228731669864 \cdot {x}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto 1.6316775383 + \color{blue}{\left(1.3436228731669864 \cdot {x}^{2} + -2.134856267379707 \cdot x\right)} \]

      2. unpow2 99.0%

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

      3. associate-*r* 99.0%

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

      4. distribute-rgt-out 99.0%

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

      5. *-commutative 99.0%

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

   8. Simplified 99.0%

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

   if 1.5600000000000001 < 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 99.3%

      \[\leadsto \color{blue}{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. fma-def 0.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.70711, x, 4.2702753202410175 \cdot \frac{1}{x}\right)}} \cdot \sqrt{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]

      3. un-div-inv 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \color{blue}{\frac{4.2702753202410175}{x}}\right)} \cdot \sqrt{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]

      4. fma-def 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.70711, x, 4.2702753202410175 \cdot \frac{1}{x}\right)}} \]

      5. un-div-inv 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\mathsf{fma}\left(-0.70711, x, \color{blue}{\frac{4.2702753202410175}{x}}\right)} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

      2. fma-udef 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 1.56:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 + -2.134856267379707\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 2.8)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 2.8d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 2.8)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 2.8):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 2.8))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 2.8)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 2.8]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 2.7999999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.4%

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

   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. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (if (<= x 2.8)
     (+ 1.6316775383 (* x -2.134856267379707))
     (+ (* x -0.70711) (/ 4.2702753202410175 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else if (x <= 2.8d0) then
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    else
        tmp = (x * (-0.70711d0)) + (4.2702753202410175d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else if (x <= 2.8) {
		tmp = 1.6316775383 + (x * -2.134856267379707);
	} else {
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	elif x <= 2.8:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	else:
		tmp = (x * -0.70711) + (4.2702753202410175 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	elseif (x <= 2.8)
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	else
		tmp = Float64(Float64(x * -0.70711) + Float64(4.2702753202410175 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	elseif (x <= 2.8)
		tmp = 1.6316775383 + (x * -2.134856267379707);
	else
		tmp = (x * -0.70711) + (4.2702753202410175 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.8], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision], N[(N[(x * -0.70711), $MachinePrecision] + N[(4.2702753202410175 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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. Taylor expanded in x around inf 97.4%

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

   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. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   4. Simplified 98.2%

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

   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 99.3%

      \[\leadsto \color{blue}{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

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

      2. fma-def 0.0%

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.70711, x, 4.2702753202410175 \cdot \frac{1}{x}\right)}} \cdot \sqrt{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]

      3. un-div-inv 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \color{blue}{\frac{4.2702753202410175}{x}}\right)} \cdot \sqrt{-0.70711 \cdot x + 4.2702753202410175 \cdot \frac{1}{x}} \]

      4. fma-def 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-0.70711, x, 4.2702753202410175 \cdot \frac{1}{x}\right)}} \]

      5. un-div-inv 0.0%

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\mathsf{fma}\left(-0.70711, x, \color{blue}{\frac{4.2702753202410175}{x}}\right)} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \cdot \sqrt{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.3%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.70711, x, \frac{4.2702753202410175}{x}\right)} \]

      2. fma-udef 99.3%

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

      3. *-commutative 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{elif}\;x \leq 2.8:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711 + \frac{4.2702753202410175}{x}\\ \end{array} \]

Alternative 7: 98.8% 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 99.7%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   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. Taylor expanded in x around 0 98.2%

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

      1. *-commutative 98.2%

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

   4. Simplified 98.2%

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

4. Final simplification 98.0%

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

Alternative 8: 98.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05) (* x -0.70711) (if (<= x 1.2) 1.6316775383 (* x -0.70711))))

C

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

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = x * -0.70711
	elif x <= 1.2:
		tmp = 1.6316775383
	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 <= 1.2)
		tmp = 1.6316775383;
	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 <= 1.2)
		tmp = 1.6316775383;
	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, 1.2], 1.6316775383, N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.19999999999999996 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. *-commutative 97.8%

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

   4. Simplified 97.8%

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

   if -1.05000000000000004 < x < 1.19999999999999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;1.6316775383\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.70711\\ \end{array} \]

Alternative 9: 49.5% 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. Taylor expanded in x around 0 44.1%

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

3. Final simplification 44.1%

   \[\leadsto 1.6316775383 \]

Reproduce

herbie shell --seed 2023271 
(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)))