Given's Rotation SVD example, simplified

Average Error: 15.3 → 0.0

Time: 14.4s

Precision: binary64

Cost: 45892

Math

\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

↓

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))

↓

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.2)
   (fma
    x
    (* x 0.125)
    (fma
     0.0673828125
     (pow x 6.0)
     (fma -0.0859375 (pow x 4.0) (* -0.056243896484375 (pow x 8.0)))))
   (/
    (+ 0.5 (/ -0.5 (hypot 1.0 x)))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))

C

double code(double x) {
	return 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
}

↓

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.2) {
		tmp = fma(x, (x * 0.125), fma(0.0673828125, pow(x, 6.0), fma(-0.0859375, pow(x, 4.0), (-0.056243896484375 * pow(x, 8.0)))));
	} else {
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	}
	return tmp;
}

Julia

function code(x)
	return Float64(1.0 - sqrt(Float64(0.5 * Float64(1.0 + Float64(1.0 / hypot(1.0, x))))))
end

↓

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.2)
		tmp = fma(x, Float64(x * 0.125), fma(0.0673828125, (x ^ 6.0), fma(-0.0859375, (x ^ 4.0), Float64(-0.056243896484375 * (x ^ 8.0)))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / hypot(1.0, x))) / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x))))));
	end
	return tmp
end

Wolfram

code[x_] := N[(1.0 - N[Sqrt[N[(0.5 * N[(1.0 + N[(1.0 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.2], N[(x * N[(x * 0.125), $MachinePrecision] + N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision] + N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.19999999999999996

   1. Initial program 29.4

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Simplified 29.4

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start] 29.4	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 29.4	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 29.4	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 29.4	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 29.4	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around 0 0.1

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)} \]

   4. Applied egg-rr 29.5

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.0673828125 \cdot {x}^{6}\right)} - \left(1 - \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right)\right)} \]

   5. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)\right)\right)} \]
      Proof

      [Start] 29.5	\[ e^{\mathsf{log1p}\left(0.0673828125 \cdot {x}^{6}\right)} - \left(1 - \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right)\right) \]
      associate--r- [=>] 0.2	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(0.0673828125 \cdot {x}^{6}\right)} - 1\right) + \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right)} \]
      expm1-def [=>] 0.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.0673828125 \cdot {x}^{6}\right)\right)} + \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right) \]
      expm1-log1p [=>] 0.1	\[ \color{blue}{0.0673828125 \cdot {x}^{6}} + \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right) \]
      associate-+r+ [=>] 0.1	\[ \color{blue}{\left(0.0673828125 \cdot {x}^{6} + \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right) + x \cdot \left(x \cdot 0.125\right)} \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)} + x \cdot \left(x \cdot 0.125\right) \]
      +-commutative [=>] 0.1	\[ \color{blue}{x \cdot \left(x \cdot 0.125\right) + \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)} \]
      fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)\right)} \]
      fma-def [<=] 0.1	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}}\right)\right) \]
      +-commutative [=>] 0.1	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{-0.0859375 \cdot {x}^{4} + -0.056243896484375 \cdot {x}^{8}}\right)\right) \]
      fma-def [=>] 0.1	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{\mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)}\right)\right) \]

   if 1.19999999999999996 < (hypot.f64 1 x)

   1. Initial program 1.0

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]

   2. Simplified 1.0

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      Proof

      [Start] 1.0	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 1.0	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 1.0	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 1.0	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 1.0	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 1.0

      \[\leadsto \color{blue}{\frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      Proof

      [Start] 1.0	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      sqr-neg [=>] 1.0	\[ \frac{1 - \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      rem-square-sqrt [=>] 0.0	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 0.0	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 0.0	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	33412

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	33348

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;0.0673828125 \cdot {x}^{6} + \left(\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right) + x \cdot \left(x \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	32900

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, -0.0859375 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 4
Error	0.1
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.2:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 5
Error	0.4
Cost	20360

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.012:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6
Error	0.5
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00002:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 7
Error	0.5
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.00002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot 0.125, x, -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 8
Error	0.6
Cost	7624

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{\frac{\frac{0.25}{x \cdot x} + -0.25}{-0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternative 9
Error	0.6
Cost	7364

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{t_0}\\ \end{array} \]

Alternative 10
Error	0.8
Cost	7240

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{-0.5}{x}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \]

Alternative 11
Error	1.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \]

Alternative 12
Error	0.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{-0.5}{x}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{x}}\\ \end{array} \]

Alternative 13
Error	1.3
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 14
Error	26.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;0.125\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \]

Alternative 15
Error	26.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -0.66:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{2}\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \]

Alternative 16
Error	26.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0.125\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \]

Alternative 17
Error	40.9
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-77}:\\ \;\;\;\;0.125\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.125\\ \end{array} \]

Alternative 18
Error	46.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023237 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))