Given's Rotation SVD example, simplified

Average Accuracy: 75.8% → 99.8%

Time: 13.6s

Precision: binary64

Cost: 26756

Math

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

↓

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

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) 2.0)
   (+
    (+ (* 0.125 (* x x)) (* 0.0673828125 (pow x 6.0)))
    (* -0.0859375 (pow x 4.0)))
   (/
    (+ 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) <= 2.0) {
		tmp = ((0.125 * (x * x)) + (0.0673828125 * pow(x, 6.0))) + (-0.0859375 * pow(x, 4.0));
	} else {
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = ((0.125 * (x * x)) + (0.0673828125 * Math.pow(x, 6.0))) + (-0.0859375 * Math.pow(x, 4.0));
	} else {
		tmp = (0.5 + (-0.5 / Math.hypot(1.0, x))) / (1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x)))));
	}
	return tmp;
}

Python

def code(x):
	return 1.0 - math.sqrt((0.5 * (1.0 + (1.0 / math.hypot(1.0, x)))))

↓

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = ((0.125 * (x * x)) + (0.0673828125 * math.pow(x, 6.0))) + (-0.0859375 * math.pow(x, 4.0))
	else:
		tmp = (0.5 + (-0.5 / math.hypot(1.0, x))) / (1.0 + math.sqrt((0.5 + (0.5 / math.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) <= 2.0)
		tmp = Float64(Float64(Float64(0.125 * Float64(x * x)) + Float64(0.0673828125 * (x ^ 6.0))) + Float64(-0.0859375 * (x ^ 4.0)));
	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

MATLAB

function tmp = code(x)
	tmp = 1.0 - sqrt((0.5 * (1.0 + (1.0 / hypot(1.0, x)))));
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = ((0.125 * (x * x)) + (0.0673828125 * (x ^ 6.0))) + (-0.0859375 * (x ^ 4.0));
	else
		tmp = (0.5 + (-0.5 / hypot(1.0, x))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	end
	tmp_2 = 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], 2.0], N[(N[(N[(0.125 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $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) < 2

   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 99.7%

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

   4. Applied egg-rr 99.0%

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

      [Start] 99.7	\[ 0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      add-sqr-sqrt [=>] 99.3	\[ \color{blue}{\sqrt{0.125 \cdot {x}^{2}} \cdot \sqrt{0.125 \cdot {x}^{2}}} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      fma-def [=>] 99.3	\[ \color{blue}{\mathsf{fma}\left(\sqrt{0.125 \cdot {x}^{2}}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)} \]
      *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\sqrt{\color{blue}{{x}^{2} \cdot 0.125}}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      sqrt-prod [=>] 99.3	\[ \mathsf{fma}\left(\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{0.125}}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      sqrt-pow1 [=>] 74.1	\[ \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{0.125}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      metadata-eval [=>] 74.1	\[ \mathsf{fma}\left({x}^{\color{blue}{1}} \cdot \sqrt{0.125}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      pow1 [<=] 74.1	\[ \mathsf{fma}\left(\color{blue}{x} \cdot \sqrt{0.125}, \sqrt{0.125 \cdot {x}^{2}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      *-commutative [=>] 74.1	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, \sqrt{\color{blue}{{x}^{2} \cdot 0.125}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      sqrt-prod [=>] 74.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, \color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{0.125}}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      sqrt-pow1 [=>] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, \color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{0.125}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      metadata-eval [=>] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, {x}^{\color{blue}{1}} \cdot \sqrt{0.125}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      pow1 [<=] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, \color{blue}{x} \cdot \sqrt{0.125}, 0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right) \]
      +-commutative [=>] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, x \cdot \sqrt{0.125}, \color{blue}{-0.0859375 \cdot {x}^{4} + 0.0673828125 \cdot {x}^{6}}\right) \]
      fma-def [=>] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, x \cdot \sqrt{0.125}, \color{blue}{\mathsf{fma}\left(-0.0859375, {x}^{4}, 0.0673828125 \cdot {x}^{6}\right)}\right) \]

   5. Applied egg-rr 99.7%

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

      [Start] 99.0	\[ \mathsf{fma}\left(x \cdot \sqrt{0.125}, x \cdot \sqrt{0.125}, \mathsf{fma}\left(-0.0859375, {x}^{4}, 0.0673828125 \cdot {x}^{6}\right)\right) \]
      fma-udef [=>] 99.0	\[ \color{blue}{\left(x \cdot \sqrt{0.125}\right) \cdot \left(x \cdot \sqrt{0.125}\right) + \mathsf{fma}\left(-0.0859375, {x}^{4}, 0.0673828125 \cdot {x}^{6}\right)} \]
      fma-udef [=>] 99.0	\[ \left(x \cdot \sqrt{0.125}\right) \cdot \left(x \cdot \sqrt{0.125}\right) + \color{blue}{\left(-0.0859375 \cdot {x}^{4} + 0.0673828125 \cdot {x}^{6}\right)} \]
      +-commutative [=>] 99.0	\[ \left(x \cdot \sqrt{0.125}\right) \cdot \left(x \cdot \sqrt{0.125}\right) + \color{blue}{\left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)} \]
      associate-+r+ [=>] 99.0	\[ \color{blue}{\left(\left(x \cdot \sqrt{0.125}\right) \cdot \left(x \cdot \sqrt{0.125}\right) + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4}} \]
      *-commutative [=>] 99.0	\[ \left(\color{blue}{\left(\sqrt{0.125} \cdot x\right)} \cdot \left(x \cdot \sqrt{0.125}\right) + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4} \]
      *-commutative [=>] 99.0	\[ \left(\left(\sqrt{0.125} \cdot x\right) \cdot \color{blue}{\left(\sqrt{0.125} \cdot x\right)} + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4} \]
      swap-sqr [=>] 98.7	\[ \left(\color{blue}{\left(\sqrt{0.125} \cdot \sqrt{0.125}\right) \cdot \left(x \cdot x\right)} + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4} \]
      add-sqr-sqrt [<=] 99.7	\[ \left(\color{blue}{0.125} \cdot \left(x \cdot x\right) + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4} \]

   if 2 < (hypot.f64 1 x)

   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. Applied egg-rr 98.5%

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

      [Start] 98.5	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      sub-neg [=>] 98.5	\[ \color{blue}{1 + \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \]
      flip-+ [=>] 98.5	\[ \color{blue}{\frac{1 \cdot 1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \]
      metadata-eval [=>] 98.5	\[ \frac{\color{blue}{1} - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \]
      neg-sub0 [=>] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 - \color{blue}{\left(0 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}} \]
      metadata-eval [<=] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 - \left(\color{blue}{\log 1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \]
      associate--r- [=>] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{\color{blue}{\left(1 - \log 1\right) + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      metadata-eval [=>] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{\left(1 - \color{blue}{0}\right) + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{\color{blue}{1} + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   4. Simplified 100.0%

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

      [Start] 98.5	\[ \frac{1 - \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \left(-\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      sqr-neg [=>] 98.5	\[ \frac{1 - \color{blue}{\sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      rem-square-sqrt [=>] 100.0	\[ \frac{1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      associate--r+ [=>] 100.0	\[ \frac{\color{blue}{\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 100.0	\[ \frac{\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\left(0.125 \cdot \left(x \cdot x\right) + 0.0673828125 \cdot {x}^{6}\right) + -0.0859375 \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{-0.5}{x}}} \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + x \cdot \left(x \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 3
Accuracy	98.7%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1 \lor \neg \left(x \leq 1.1\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + x \cdot \left(x \cdot 0.125\right)\\ \end{array} \]

Alternative 4
Accuracy	98.5%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 5
Accuracy	97.7%
Cost	6857

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \lor \neg \left(x \leq 1.55\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 6
Accuracy	60.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \end{array} \]

Alternative 7
Accuracy	60.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;0.5 \cdot \left(0.5 + \frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.75:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(0.5 + \frac{0.5}{x}\right)\\ \end{array} \]

Alternative 8
Accuracy	60.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 9
Accuracy	49.7%
Cost	320

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

Alternative 10
Accuracy	26.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))