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

Percentage Accurate: 99.9% → 99.9%

Time: 11.9s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% 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, 0.1× 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 99.9%

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

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

   4. distribute-rgt-neg-out 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-define 99.9%

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

   8. metadata-eval 99.9%

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

   9. associate-*r/ 99.9%

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

   10. +-commutative 99.9%

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

   11. distribute-lft-in 99.9%

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

   12. *-commutative 99.9%

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

   13. associate-*r* 99.9%

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

   14. *-commutative 99.9%

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

   15. fma-define 99.9%

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

   16. metadata-eval 99.9%

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

   17. metadata-eval 99.9%

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

   18. +-commutative 99.9%

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

   19. fma-define 99.9%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

8. Applied egg-rr 99.9%

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

Alternative 2: 99.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-undefine 99.9%

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

   6. fma-undefine 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

6. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

8. Applied egg-rr 99.9%

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

   1. fma-undefine 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 1.6)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x (- (* x 1.3436228731669864) 2.134856267379707)))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.6)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.6d0))) then
        tmp = 0.70711d0 * ((6.039053782637804d0 / x) - x)
    else
        tmp = 1.6316775383d0 + (x * ((x * 1.3436228731669864d0) - 2.134856267379707d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(x <= 1.6)) {
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	} else {
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.6):
		tmp = 0.70711 * ((6.039053782637804 / x) - x)
	else:
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.6))
		tmp = Float64(0.70711 * Float64(Float64(6.039053782637804 / x) - x));
	else
		tmp = Float64(1.6316775383 + Float64(x * Float64(Float64(x * 1.3436228731669864) - 2.134856267379707)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.6)))
		tmp = 0.70711 * ((6.039053782637804 / x) - x);
	else
		tmp = 1.6316775383 + (x * ((x * 1.3436228731669864) - 2.134856267379707));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(0.70711 * N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(1.6316775383 + N[(x * N[(N[(x * 1.3436228731669864), $MachinePrecision] - 2.134856267379707), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.6000000000000001 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 98.9%

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

   if -1.1000000000000001 < x < 1.6000000000000001

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. associate-*r/ 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. *-commutative 100.0%

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

      15. fma-define 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-commutative 100.0%

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

      19. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot \left(x \cdot 1.3436228731669864 - 2.134856267379707\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.1) (not (<= x 2.8)))
   (* 0.70711 (- (/ 6.039053782637804 x) x))
   (+ 1.6316775383 (* x -2.134856267379707))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.1) || !(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 <= (-1.1d0)) .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 <= -1.1) || !(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 <= -1.1) 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 <= -1.1) || !(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 <= -1.1) || ~((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, -1.1], 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 < -1.1000000000000001 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. Add Preprocessing

   3. Taylor expanded in x around inf 98.9%

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

   if -1.1000000000000001 < x < 2.7999999999999998

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. associate-*r/ 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. *-commutative 100.0%

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

      15. fma-define 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-commutative 100.0%

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

      19. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 2.8\right):\\ \;\;\;\;0.70711 \cdot \left(\frac{6.039053782637804}{x} - x\right)\\ \mathbf{else}:\\ \;\;\;\;1.6316775383 + x \cdot -2.134856267379707\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.1d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0 + (x * (-2.134856267379707d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if (x <= -1.1) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383 + (x * -2.134856267379707)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.1) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = Float64(1.6316775383 + Float64(x * -2.134856267379707));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.1) || ~((x <= 1.15)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383 + (x * -2.134856267379707);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.1], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], N[(1.6316775383 + N[(x * -2.134856267379707), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1000000000000001 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-lft-in 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-define 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. +-commutative 99.7%

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

      19. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -1.1000000000000001 < x < 1.1499999999999999

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-lft-in 100.0%

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

      4. distribute-rgt-neg-out 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. metadata-eval 100.0%

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

      9. associate-*r/ 100.0%

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

      10. +-commutative 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. *-commutative 100.0%

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

      13. associate-*r* 100.0%

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

      14. *-commutative 100.0%

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

      15. fma-define 100.0%

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

      16. metadata-eval 100.0%

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

      17. metadata-eval 100.0%

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

      18. +-commutative 100.0%

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

      19. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.8%

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

Alternative 7: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (/
    1.0
    (+
     0.4333638132548656
     (* x (+ 0.37920088514346545 (* x -0.025050834237766436)))))
   x)))

C

double code(double x) {
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * ((1.0d0 / (0.4333638132548656d0 + (x * (0.37920088514346545d0 + (x * (-0.025050834237766436d0)))))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x);
}

Python

def code(x):
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * Float64(0.37920088514346545 + Float64(x * -0.025050834237766436))))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * ((1.0 / (0.4333638132548656 + (x * (0.37920088514346545 + (x * -0.025050834237766436))))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(1.0 / N[(0.4333638132548656 + N[(x * N[(0.37920088514346545 + N[(x * -0.025050834237766436), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-undefine 99.9%

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

   6. fma-undefine 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 98.9%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + -0.025050834237766436 \cdot x\right)}} - x\right) \]
8. Step-by-step derivation

   1. *-commutative 98.9%

      \[\leadsto 0.70711 \cdot \left(\frac{1}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + \color{blue}{x \cdot -0.025050834237766436}\right)} - x\right) \]

9. Simplified 98.9%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{0.4333638132548656 + x \cdot \left(0.37920088514346545 + x \cdot -0.025050834237766436\right)}} - x\right) \]
10. Add Preprocessing

Alternative 8: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -3.5) (not (<= x 1.15))) (* x -0.70711) 1.6316775383))

C

double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-3.5d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = x * (-0.70711d0)
    else
        tmp = 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -3.5) || !(x <= 1.15)) {
		tmp = x * -0.70711;
	} else {
		tmp = 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -3.5) or not (x <= 1.15):
		tmp = x * -0.70711
	else:
		tmp = 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -3.5) || !(x <= 1.15))
		tmp = Float64(x * -0.70711);
	else
		tmp = 1.6316775383;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -3.5) || ~((x <= 1.15)))
		tmp = x * -0.70711;
	else
		tmp = 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -3.5], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], N[(x * -0.70711), $MachinePrecision], 1.6316775383]

Derivation

1. Split input into 2 regimes

2. if x < -3.5 or 1.1499999999999999 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-lft-in 99.7%

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

      4. distribute-rgt-neg-out 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.7%

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

      8. metadata-eval 99.7%

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

      9. associate-*r/ 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-lft-in 99.7%

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

      12. *-commutative 99.7%

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

      13. associate-*r* 99.7%

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

      14. *-commutative 99.7%

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

      15. fma-define 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

      18. +-commutative 99.7%

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

      19. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -3.5 < x < 1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 99.0%

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

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. flip-- 99.0%

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

      3. associate-*l/ 99.0%

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

      4. pow2 99.0%

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

      5. +-commutative 99.0%

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

      6. fma-undefine 99.0%

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

      7. +-commutative 99.0%

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

      8. fma-define 99.0%

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

      9. pow2 99.0%

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

      10. +-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around 0 98.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;1.6316775383\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{1}{0.4333638132548656 + x \cdot 0.37920088514346545} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* 0.70711 (- (/ 1.0 (+ 0.4333638132548656 (* x 0.37920088514346545))) x)))

C

double code(double x) {
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.70711d0 * ((1.0d0 / (0.4333638132548656d0 + (x * 0.37920088514346545d0))) - x)
end function

Java

public static double code(double x) {
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x);
}

Python

def code(x):
	return 0.70711 * ((1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x)

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(1.0 / Float64(0.4333638132548656 + Float64(x * 0.37920088514346545))) - x))
end

MATLAB

function tmp = code(x)
	tmp = 0.70711 * ((1.0 / (0.4333638132548656 + (x * 0.37920088514346545))) - x);
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(1.0 / N[(0.4333638132548656 + N[(x * 0.37920088514346545), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-undefine 99.9%

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

   6. fma-undefine 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around 0 98.8%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{0.4333638132548656 + 0.37920088514346545 \cdot x}} - x\right) \]
8. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto 0.70711 \cdot \left(\frac{1}{0.4333638132548656 + \color{blue}{x \cdot 0.37920088514346545}} - x\right) \]

9. Simplified 98.8%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{0.4333638132548656 + x \cdot 0.37920088514346545}} - x\right) \]
10. Add Preprocessing

Alternative 10: 50.8% accurate, 19.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.6316775383)

C

double code(double x) {
	return 1.6316775383;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.6316775383

Julia

function code(x)
	return 1.6316775383
end

MATLAB

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

Wolfram

code[x_] := 1.6316775383

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

5. Simplified 98.6%

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

   1. *-commutative 98.6%

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

   2. flip-- 77.3%

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

   3. associate-*l/ 77.3%

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

   4. pow2 77.3%

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

   5. +-commutative 77.3%

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

   6. fma-undefine 77.3%

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

   7. +-commutative 77.3%

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

   8. fma-define 77.3%

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

   9. pow2 77.3%

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

   10. +-commutative 77.3%

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

7. Applied egg-rr 77.3%

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

8. Taylor expanded in x around 0 54.9%

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

Alternative 11: 10.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 0.3135931908666891 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.3135931908666891)

C

double code(double x) {
	return 0.3135931908666891;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3135931908666891d0
end function

Java

public static double code(double x) {
	return 0.3135931908666891;
}

Python

def code(x):
	return 0.3135931908666891

Julia

function code(x)
	return 0.3135931908666891
end

MATLAB

function tmp = code(x)
	tmp = 0.3135931908666891;
end

Wolfram

code[x_] := 0.3135931908666891

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. +-commutative 99.9%

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

   4. +-commutative 99.9%

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

   5. fma-undefine 99.9%

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

   6. fma-undefine 99.9%

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

   7. +-commutative 99.9%

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

   8. fma-define 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

6. Applied egg-rr 99.9%

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

7. Taylor expanded in x around inf 54.3%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{x \cdot \left(0.16558885480950444 + 2.254864010426164 \cdot \frac{1}{x}\right)}} - x\right) \]
8. Step-by-step derivation

   1. associate-*r/ 54.3%

      \[\leadsto 0.70711 \cdot \left(\frac{1}{x \cdot \left(0.16558885480950444 + \color{blue}{\frac{2.254864010426164 \cdot 1}{x}}\right)} - x\right) \]

   2. metadata-eval 54.3%

      \[\leadsto 0.70711 \cdot \left(\frac{1}{x \cdot \left(0.16558885480950444 + \frac{\color{blue}{2.254864010426164}}{x}\right)} - x\right) \]

9. Simplified 54.3%

   \[\leadsto 0.70711 \cdot \left(\frac{1}{\color{blue}{x \cdot \left(0.16558885480950444 + \frac{2.254864010426164}{x}\right)}} - x\right) \]

10. Taylor expanded in x around 0 10.5%

    \[\leadsto \color{blue}{0.3135931908666891} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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)))