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

Percentage Accurate: 99.8% → 99.8%

Time: 6.6s

Alternatives: 8

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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, 0.27061 \cdot x\right) + -1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.70711
  (-
   (/
    (+
     2.30753
     (+ (fma 3.695354938841876 (/ (* 0.27061 x) x) (* 0.27061 x)) -1.0))
    (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
   x)))

C

double code(double x) {
	return 0.70711 * (((2.30753 + (fma(3.695354938841876, ((0.27061 * x) / x), (0.27061 * x)) + -1.0)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x);
}

Julia

function code(x)
	return Float64(0.70711 * Float64(Float64(Float64(2.30753 + Float64(fma(3.695354938841876, Float64(Float64(0.27061 * x) / x), Float64(0.27061 * x)) + -1.0)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x))
end

Wolfram

code[x_] := N[(0.70711 * N[(N[(N[(2.30753 + N[(N[(3.695354938841876 * N[(N[(0.27061 * x), $MachinePrecision] / x), $MachinePrecision] + N[(0.27061 * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 74.9%

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

   2. expm1-udef 74.9%

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

3. Applied egg-rr 74.9%

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

4. Taylor expanded in x around inf 49.0%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\color{blue}{\left(3.695354938841876 \cdot \frac{e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}}{x} + e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right)} - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]
5. Step-by-step derivation

   1. fma-def 49.0%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\color{blue}{\mathsf{fma}\left(3.695354938841876, \frac{e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right)} - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   2. exp-sum 49.0%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{\color{blue}{e^{\log 0.27061} \cdot e^{-1 \cdot \log \left(\frac{1}{x}\right)}}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   3. mul-1-neg 49.0%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{e^{\log 0.27061} \cdot e^{\color{blue}{-\log \left(\frac{1}{x}\right)}}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   4. log-rec 49.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{e^{\log 0.27061} \cdot e^{-\color{blue}{\left(-\log x\right)}}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   5. remove-double-neg 49.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{e^{\log 0.27061} \cdot e^{\color{blue}{\log x}}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   6. rem-exp-log 49.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{\color{blue}{0.27061} \cdot e^{\log x}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   7. rem-exp-log 51.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot \color{blue}{x}}{x}, e^{\log 0.27061 + -1 \cdot \log \left(\frac{1}{x}\right)}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   8. exp-sum 51.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, \color{blue}{e^{\log 0.27061} \cdot e^{-1 \cdot \log \left(\frac{1}{x}\right)}}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   9. mul-1-neg 51.1%

      \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, e^{\log 0.27061} \cdot e^{\color{blue}{-\log \left(\frac{1}{x}\right)}}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   10. log-rec 51.1%

       \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, e^{\log 0.27061} \cdot e^{-\color{blue}{\left(-\log x\right)}}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   11. remove-double-neg 51.1%

       \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, e^{\log 0.27061} \cdot e^{\color{blue}{\log x}}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   12. rem-exp-log 51.1%

       \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, \color{blue}{0.27061} \cdot e^{\log x}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

   13. rem-exp-log 99.9%

       \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, 0.27061 \cdot \color{blue}{x}\right) - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

6. Simplified 99.9%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\color{blue}{\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, 0.27061 \cdot x\right)} - 1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

7. Final simplification 99.9%

   \[\leadsto 0.70711 \cdot \left(\frac{2.30753 + \left(\mathsf{fma}\left(3.695354938841876, \frac{0.27061 \cdot x}{x}, 0.27061 \cdot x\right) + -1\right)}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\right) \]

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 3.6:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{58.14938538768042}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (if (<= x 3.6)
     (+ (* x -2.134856267379707) 1.6316775383)
     (+
      (/ 4.2702753202410175 x)
      (- (* x -0.70711) (/ 58.14938538768042 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 3.6) {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	} else {
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else if (x <= 3.6d0) then
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    else
        tmp = (4.2702753202410175d0 / x) + ((x * (-0.70711d0)) - (58.14938538768042d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else if (x <= 3.6) {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	} else {
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	elif x <= 3.6:
		tmp = (x * -2.134856267379707) + 1.6316775383
	else:
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	elseif (x <= 3.6)
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	else
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(Float64(x * -0.70711) - Float64(58.14938538768042 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	elseif (x <= 3.6)
		tmp = (x * -2.134856267379707) + 1.6316775383;
	else
		tmp = (4.2702753202410175 / x) + ((x * -0.70711) - (58.14938538768042 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(N[(x * -0.70711), $MachinePrecision] - N[(58.14938538768042 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if -1.05000000000000004 < x < 3.60000000000000009

   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/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

   if 3.60000000000000009 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.8%

      \[\leadsto \color{blue}{\left(4.2702753202410175 \cdot \frac{1}{x} + -0.70711 \cdot x\right) - 58.14938538768042 \cdot \frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{4.2702753202410175 \cdot \frac{1}{x} + \left(-0.70711 \cdot x - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right)} \]

      2. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{4.2702753202410175 \cdot 1}{x}} + \left(-0.70711 \cdot x - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{4.2702753202410175}}{x} + \left(-0.70711 \cdot x - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \frac{4.2702753202410175}{x} + \left(\color{blue}{x \cdot -0.70711} - 58.14938538768042 \cdot \frac{1}{{x}^{2}}\right) \]

      5. unpow2 99.8%

         \[\leadsto \frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - 58.14938538768042 \cdot \frac{1}{\color{blue}{x \cdot x}}\right) \]

      6. associate-*r/ 99.8%

         \[\leadsto \frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \color{blue}{\frac{58.14938538768042 \cdot 1}{x \cdot x}}\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{\color{blue}{58.14938538768042}}{x \cdot x}\right) \]

   6. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{58.14938538768042}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{elif}\;x \leq 3.6:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \mathbf{else}:\\ \;\;\;\;\frac{4.2702753202410175}{x} + \left(x \cdot -0.70711 - \frac{58.14938538768042}{x \cdot x}\right)\\ \end{array} \]

Alternative 4: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 98.2%

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

   1. *-commutative 98.2%

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

4. Simplified 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 0.75)))
   (+ (/ 4.2702753202410175 x) (* x -0.70711))
   (+ (* x -2.134856267379707) 1.6316775383)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = (4.2702753202410175d0 / x) + (x * (-0.70711d0))
    else
        tmp = (x * (-2.134856267379707d0)) + 1.6316775383d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 0.75)) {
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	} else {
		tmp = (x * -2.134856267379707) + 1.6316775383;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 0.75):
		tmp = (4.2702753202410175 / x) + (x * -0.70711)
	else:
		tmp = (x * -2.134856267379707) + 1.6316775383
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 0.75))
		tmp = Float64(Float64(4.2702753202410175 / x) + Float64(x * -0.70711));
	else
		tmp = Float64(Float64(x * -2.134856267379707) + 1.6316775383);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 0.75)))
		tmp = (4.2702753202410175 / x) + (x * -0.70711);
	else
		tmp = (x * -2.134856267379707) + 1.6316775383;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[(N[(4.2702753202410175 / x), $MachinePrecision] + N[(x * -0.70711), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 0.75 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if -1.05000000000000004 < x < 0.75

   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/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{4.2702753202410175}{x} + x \cdot -0.70711\\ \mathbf{else}:\\ \;\;\;\;x \cdot -2.134856267379707 + 1.6316775383\\ \end{array} \]

Alternative 6: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = x * -0.70711;
	} else if (x <= 1.15) {
		tmp = (x * -2.134856267379707) + 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 <= 1.15d0) then
        tmp = (x * (-2.134856267379707d0)) + 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 <= 1.15) {
		tmp = (x * -2.134856267379707) + 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 <= 1.15:
		tmp = (x * -2.134856267379707) + 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 <= 1.15)
		tmp = Float64(Float64(x * -2.134856267379707) + 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 <= 1.15)
		tmp = (x * -2.134856267379707) + 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, 1.15], N[(N[(x * -2.134856267379707), $MachinePrecision] + 1.6316775383), $MachinePrecision], N[(x * -0.70711), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. fma-def 99.8%

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

      9. metadata-eval 99.8%

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

      10. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-rgt-in 99.9%

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

      4. distribute-lft-neg-out 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.5%

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

4. Final simplification 98.4%

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

Alternative 7: 10.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x \cdot -2.134856267379707 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -2.134856267379707))

C

double code(double x) {
	return x * -2.134856267379707;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-2.134856267379707d0)
end function

Java

public static double code(double x) {
	return x * -2.134856267379707;
}

Python

def code(x):
	return x * -2.134856267379707

Julia

function code(x)
	return Float64(x * -2.134856267379707)
end

MATLAB

function tmp = code(x)
	tmp = x * -2.134856267379707;
end

Wolfram

code[x_] := N[(x * -2.134856267379707), $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-rgt-in 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 58.5%

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

5. Taylor expanded in x around inf 10.6%

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

   1. *-commutative 10.6%

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

7. Simplified 10.6%

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

8. Final simplification 10.6%

   \[\leadsto x \cdot -2.134856267379707 \]

Alternative 8: 51.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.70711 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.70711))

C

double code(double x) {
	return x * -0.70711;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.70711d0)
end function

Java

public static double code(double x) {
	return x * -0.70711;
}

Python

def code(x):
	return x * -0.70711

Julia

function code(x)
	return Float64(x * -0.70711)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.70711;
end

Wolfram

code[x_] := N[(x * -0.70711), $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-rgt-in 99.9%

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

   4. distribute-lft-neg-out 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 50.4%

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

   1. *-commutative 50.4%

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

6. Simplified 50.4%

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

7. Final simplification 50.4%

   \[\leadsto x \cdot -0.70711 \]

Reproduce

herbie shell --seed 2023192 
(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)))