Given's Rotation SVD example, simplified

Average Accuracy: 75.6% → 99.7%

Time: 14.7s

Precision: binary64

Cost: 39560

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.027:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{t_0 \cdot \left(1 + \sqrt{t_0}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= x -1.1)
     (/ (+ 0.5 (/ 0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))
     (if (<= x 0.027)
       (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)))))
       (/ (+ 0.25 (/ -0.25 (+ 1.0 (* x x)))) (* t_0 (+ 1.0 (sqrt t_0))))))))

C

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

↓

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (x <= -1.1) {
		tmp = (0.5 + (0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	} else if (x <= 0.027) {
		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 = (0.25 + (-0.25 / (1.0 + (x * x)))) / (t_0 * (1.0 + sqrt(t_0)));
	}
	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)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (x <= -1.1)
		tmp = Float64(Float64(0.5 + Float64(0.5 / x)) / Float64(1.0 + sqrt(Float64(0.5 + Float64(-0.5 / x)))));
	elseif (x <= 0.027)
		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(0.25 + Float64(-0.25 / Float64(1.0 + Float64(x * x)))) / Float64(t_0 * Float64(1.0 + sqrt(t_0))));
	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_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1], N[(N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.027], 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[(0.25 + N[(-0.25 / N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 98.5%

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

   2. Simplified 98.5%

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

      [Start] 98.5	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 98.5	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.5	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 98.5	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.5	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around -inf 97.8%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - 0.5 \cdot \frac{1}{x}}} \]

   4. Simplified 97.8%

      \[\leadsto 1 - \sqrt{\color{blue}{0.5 - \frac{0.5}{x}}} \]
      Proof

      [Start] 97.8	\[ 1 - \sqrt{0.5 - 0.5 \cdot \frac{1}{x}} \]
      associate-*r/ [=>] 97.8	\[ 1 - \sqrt{0.5 - \color{blue}{\frac{0.5 \cdot 1}{x}}} \]
      metadata-eval [=>] 97.8	\[ 1 - \sqrt{0.5 - \frac{\color{blue}{0.5}}{x}} \]

   5. Applied egg-rr 97.8%

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

      [Start] 97.8	\[ 1 - \sqrt{0.5 - \frac{0.5}{x}} \]
      sub-neg [=>] 97.8	\[ \color{blue}{1 + \left(-\sqrt{0.5 - \frac{0.5}{x}}\right)} \]
      flip-+ [=>] 97.8	\[ \color{blue}{\frac{1 \cdot 1 - \left(-\sqrt{0.5 - \frac{0.5}{x}}\right) \cdot \left(-\sqrt{0.5 - \frac{0.5}{x}}\right)}{1 - \left(-\sqrt{0.5 - \frac{0.5}{x}}\right)}} \]

   6. Simplified 99.3%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}} \]
      Proof

      [Start] 97.8	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{-0.5}{x}}\right) \cdot \left(-\sqrt{0.5 + \frac{-0.5}{x}}\right)}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \]
      sqr-neg [=>] 97.8	\[ \frac{1 - \color{blue}{\sqrt{0.5 + \frac{-0.5}{x}} \cdot \sqrt{0.5 + \frac{-0.5}{x}}}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \]
      rem-square-sqrt [=>] 99.3	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{-0.5}{x}\right)}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \]
      associate--r+ [=>] 99.3	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{-0.5}{x}}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \]
      metadata-eval [=>] 99.3	\[ \frac{\color{blue}{0.5} - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \]

   if -1.1000000000000001 < x < 0.0269999999999999997

   1. Initial program 52.2%

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

   2. Simplified 52.2%

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

      [Start] 52.2	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 52.2	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 52.2	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 52.2	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 52.2	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around 0 99.8%

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

   4. Applied egg-rr 52.0%

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

      [Start] 99.8	\[ 0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right) \]
      expm1-log1p-u [=>] 99.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)\right)\right)} \]
      expm1-udef [=>] 52.0	\[ \color{blue}{e^{\mathsf{log1p}\left(0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)\right)} - 1} \]
      fma-def [=>] 52.0	\[ e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.125, {x}^{2}, 0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)}\right)} - 1 \]
      unpow2 [=>] 52.0	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.125, \color{blue}{x \cdot x}, 0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)\right)} - 1 \]
      fma-def [=>] 52.0	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.125, x \cdot x, \color{blue}{\mathsf{fma}\left(0.0673828125, {x}^{6}, -0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)}\right)\right)} - 1 \]
      fma-def [=>] 52.0	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{\mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)}\right)\right)\right)} - 1 \]

   5. Simplified 99.8%

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

      [Start] 52.0	\[ e^{\mathsf{log1p}\left(\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)\right)\right)} - 1 \]
      expm1-def [=>] 99.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)\right)\right)\right)} \]
      expm1-log1p [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(0.125, x \cdot x, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)\right)} \]
      fma-udef [=>] 99.8	\[ \color{blue}{0.125 \cdot \left(x \cdot x\right) + \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)} \]
      *-commutative [=>] 99.8	\[ \color{blue}{\left(x \cdot x\right) \cdot 0.125} + \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right) \]
      associate-*r* [<=] 99.8	\[ \color{blue}{x \cdot \left(x \cdot 0.125\right)} + \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right) \]
      fma-def [=>] 99.8	\[ \color{blue}{\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.056243896484375, {x}^{8}, -0.0859375 \cdot {x}^{4}\right)\right)\right)} \]
      fma-def [<=] 99.8	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}}\right)\right) \]
      +-commutative [=>] 99.8	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{-0.0859375 \cdot {x}^{4} + -0.056243896484375 \cdot {x}^{8}}\right)\right) \]
      fma-def [=>] 99.8	\[ \mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \color{blue}{\mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)}\right)\right) \]

   if 0.0269999999999999997 < x

   1. Initial program 98.4%

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

   2. Simplified 98.4%

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

      [Start] 98.4	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 98.4	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.4	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 98.4	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.4	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]
      Proof

      [Start] 98.4	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      flip-- [=>] 98.4	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      metadata-eval [=>] 98.4	\[ \frac{\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      add-sqr-sqrt [<=] 99.9	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 99.9	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 99.9	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      flip-- [=>] 99.9	\[ \frac{\color{blue}{\frac{0.5 \cdot 0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate-/l/ [=>] 99.9	\[ \color{blue}{\frac{0.5 \cdot 0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{0.25}{x \cdot x + 1}}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \]
      Proof

      [Start] 99.9	\[ \frac{0.25 - \frac{0.25}{1 + x \cdot x}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      /-rgt-identity [<=] 99.9	\[ \frac{\color{blue}{\frac{0.25 - \frac{0.25}{1 + x \cdot x}}{1}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      /-rgt-identity [=>] 99.9	\[ \frac{\color{blue}{0.25 - \frac{0.25}{1 + x \cdot x}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      +-commutative [=>] 99.9	\[ \frac{0.25 - \frac{0.25}{\color{blue}{x \cdot x + 1}}}{\left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      *-commutative [=>] 99.9	\[ \frac{0.25 - \frac{0.25}{x \cdot x + 1}}{\color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.027:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, \mathsf{fma}\left(-0.0859375, {x}^{4}, -0.056243896484375 \cdot {x}^{8}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	27080

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.027:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + \left(-0.056243896484375 \cdot {x}^{8} + -0.0859375 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{t_0 \cdot \left(1 + \sqrt{t_0}\right)}\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	26756

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

Alternative 3
Accuracy	99.7%
Cost	26568

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot 0.125, \mathsf{fma}\left(0.0673828125, {x}^{6}, -0.0859375 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{t_0 \cdot \left(1 + \sqrt{t_0}\right)}\\ \end{array} \]

Alternative 4
Accuracy	99.7%
Cost	21064

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 + \frac{-0.25}{1 + x \cdot x}}{t_0 \cdot \left(1 + \sqrt{t_0}\right)}\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	20360

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{t_1}}\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{t_0}} \cdot t_1\\ \end{array} \]

Alternative 6
Accuracy	99.3%
Cost	7624

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{t_1}}\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot -0.0859375, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{t_0}} \cdot t_1\\ \end{array} \]

Alternative 7
Accuracy	99.1%
Cost	7364

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

Alternative 8
Accuracy	98.7%
Cost	7240

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

Alternative 9
Accuracy	98.7%
Cost	6985

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

Alternative 10
Accuracy	98.7%
Cost	6984

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

Alternative 11
Accuracy	98.0%
Cost	6857

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

Alternative 12
Accuracy	50.3%
Cost	320

\[0.125 \cdot \left(x \cdot x\right) \]

Alternative 13
Accuracy	26.7%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023138 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))