Given's Rotation SVD example, simplified

Average Accuracy: 76.5% → 99.8%

Time: 15.2s

Precision: binary64

Cost: 33412

Math

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

↓

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

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 (pow x 2.0))
    (+
     (* 0.0673828125 (pow x 6.0))
     (+ (* -0.056243896484375 (pow x 8.0)) (* -0.0859375 (pow x 4.0)))))
   (*
    (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))
    (+ 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 * pow(x, 2.0)) + ((0.0673828125 * pow(x, 6.0)) + ((-0.056243896484375 * pow(x, 8.0)) + (-0.0859375 * pow(x, 4.0))));
	} else {
		tmp = (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))))) * (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 * Math.pow(x, 2.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + ((-0.056243896484375 * Math.pow(x, 8.0)) + (-0.0859375 * Math.pow(x, 4.0))));
	} else {
		tmp = (1.0 / (1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x)))))) * (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 * math.pow(x, 2.0)) + ((0.0673828125 * math.pow(x, 6.0)) + ((-0.056243896484375 * math.pow(x, 8.0)) + (-0.0859375 * math.pow(x, 4.0))))
	else:
		tmp = (1.0 / (1.0 + math.sqrt((0.5 + (0.5 / math.hypot(1.0, x)))))) * (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(0.125 * (x ^ 2.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(Float64(-0.056243896484375 * (x ^ 8.0)) + Float64(-0.0859375 * (x ^ 4.0)))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))))) * 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 ^ 2.0)) + ((0.0673828125 * (x ^ 6.0)) + ((-0.056243896484375 * (x ^ 8.0)) + (-0.0859375 * (x ^ 4.0))));
	else
		tmp = (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))))) * (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[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.056243896484375 * N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision] + N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.2%

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

   2. Simplified 54.2%

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

      [Start] 54.2	\[ 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      distribute-lft-in [=>] 54.2	\[ 1 - \sqrt{\color{blue}{0.5 \cdot 1 + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 54.2	\[ 1 - \sqrt{\color{blue}{0.5} + 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}} \]
      associate-*r/ [=>] 54.2	\[ 1 - \sqrt{0.5 + \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}} \]
      metadata-eval [=>] 54.2	\[ 1 - \sqrt{0.5 + \frac{\color{blue}{0.5}}{\mathsf{hypot}\left(1, x\right)}} \]

   3. Taylor expanded in x around 0 99.7%

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

   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 100.0%

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

      [Start] 98.5	\[ 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \]
      flip-- [=>] 98.5	\[ \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      div-inv [=>] 98.5	\[ \color{blue}{\left(1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
      *-commutative [=>] 98.5	\[ \color{blue}{\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(1 \cdot 1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \]
      metadata-eval [=>] 98.5	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(\color{blue}{1} - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\right) \]
      add-sqr-sqrt [<=] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(1 - \color{blue}{\left(0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}\right) \]
      associate--r+ [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \color{blue}{\left(\left(1 - 0.5\right) - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(\color{blue}{0.5} - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \]
      div-inv [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 - \color{blue}{0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}\right) \]
      cancel-sign-sub-inv [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \color{blue}{\left(0.5 + \left(-0.5\right) \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
      associate-*r/ [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \color{blue}{\frac{\left(-0.5\right) \cdot 1}{\mathsf{hypot}\left(1, x\right)}}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \frac{\color{blue}{-0.5} \cdot 1}{\mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 100.0	\[ \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	26884

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0001:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0001:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	20360

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + t_0}\\ \mathbf{elif}\;x \leq 0.0126:\\ \;\;\;\;0.125 \cdot {x}^{2} + \left(0.0673828125 \cdot {x}^{6} + -0.0859375 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	13828

\[\begin{array}{l} t_0 := \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + t_0}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \mathbf{elif}\;x \leq 0.0029:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6
Accuracy	99.4%
Cost	7624

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{t_1}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\left(2 + \sqrt{t_0}\right) + -1}\\ \end{array} \]

Alternative 7
Accuracy	99.4%
Cost	7496

\[\begin{array}{l} t_0 := 0.5 + \frac{0.5}{x}\\ t_1 := 0.5 + \frac{-0.5}{x}\\ \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\frac{t_0}{1 + \sqrt{t_1}}\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;x \cdot \left(x \cdot \mathsf{fma}\left(-0.0859375, x \cdot x, 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{1 + \sqrt{t_0}}\\ \end{array} \]

Alternative 8
Accuracy	99.0%
Cost	7364

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

Alternative 9
Accuracy	98.6%
Cost	7240

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

Alternative 10
Accuracy	98.7%
Cost	7112

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

Alternative 11
Accuracy	98.6%
Cost	7112

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

Alternative 12
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 13
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 14
Accuracy	60.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(0.125 + -0.0859375 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	60.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 1.25\right):\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 16
Accuracy	60.6%
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 17
Accuracy	3.2%
Cost	192

\[\frac{0.25}{x} \]

Alternative 18
Accuracy	13.5%
Cost	64

\[0.25 \]

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))