Given's Rotation SVD example, simplified

Percentage Accurate: 76.1% → 99.9%

Time: 14.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + t\_0 \cdot \left(t\_0 - 0.5\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (*
      (/
       (+ 0.125 (/ -0.125 (pow (hypot 1.0 x) 3.0)))
       (+ 0.25 (* t_0 (- t_0 0.5))))
      (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))))

C

double code(double x) {
	double t_0 = -0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = ((0.125 + (-0.125 / pow(hypot(1.0, x), 3.0))) / (0.25 + (t_0 * (t_0 - 0.5)))) * (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = -0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = ((0.125 + (-0.125 / Math.pow(Math.hypot(1.0, x), 3.0))) / (0.25 + (t_0 * (t_0 - 0.5)))) * (1.0 / (1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = ((0.125 + (-0.125 / math.pow(math.hypot(1.0, x), 3.0))) / (0.25 + (t_0 * (t_0 - 0.5)))) * (1.0 / (1.0 + math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))))
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(0.125 + Float64(-0.125 / (hypot(1.0, x) ^ 3.0))) / Float64(0.25 + Float64(t_0 * Float64(t_0 - 0.5)))) * Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = ((0.125 + (-0.125 / (hypot(1.0, x) ^ 3.0))) / (0.25 + (t_0 * (t_0 - 0.5)))) * (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.125 + N[(-0.125 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(t$95$0 * N[(t$95$0 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. flip3-+ 99.9%

         \[\leadsto \color{blue}{\frac{{0.5}^{3} + {\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{0.125} + {\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\color{blue}{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{0.25} + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      7. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      9. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      10. metadata-eval 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      11. distribute-neg-frac 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. cube-div 99.9%

         \[\leadsto \frac{0.125 + \color{blue}{\frac{{-0.5}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + \frac{\color{blue}{-0.125}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. distribute-rgt-out-- 99.9%

         \[\leadsto \frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0005)
   (+
    (* -0.0859375 (pow x 4.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
   (/
    (/
     (+ 0.125 (/ -0.125 (pow (hypot 1.0 x) 3.0)))
     (+ 0.25 (/ (+ 0.25 (/ 0.25 (hypot 1.0 x))) (hypot 1.0 x))))
    (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = ((0.125 + (-0.125 / pow(hypot(1.0, x), 3.0))) / (0.25 + ((0.25 + (0.25 / hypot(1.0, x))) / hypot(1.0, x)))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = ((0.125 + (-0.125 / Math.pow(Math.hypot(1.0, x), 3.0))) / (0.25 + ((0.25 + (0.25 / Math.hypot(1.0, x))) / Math.hypot(1.0, x)))) / (1.0 + Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = ((0.125 + (-0.125 / math.pow(math.hypot(1.0, x), 3.0))) / (0.25 + ((0.25 + (0.25 / math.hypot(1.0, x))) / math.hypot(1.0, x)))) / (1.0 + math.sqrt((0.5 + (0.5 / math.hypot(1.0, x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(0.125 + Float64(-0.125 / (hypot(1.0, x) ^ 3.0))) / Float64(0.25 + Float64(Float64(0.25 + Float64(0.25 / hypot(1.0, x))) / 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_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = ((0.125 + (-0.125 / (hypot(1.0, x) ^ 3.0))) / (0.25 + ((0.25 + (0.25 / hypot(1.0, x))) / hypot(1.0, x)))) / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.125 + N[(-0.125 / N[Power[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.25 + N[(N[(0.25 + N[(0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $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) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. flip3-+ 99.9%

         \[\leadsto \color{blue}{\frac{{0.5}^{3} + {\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{\color{blue}{0.125} + {\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      4. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\color{blue}{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.5 \cdot 0.5 + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{\color{blue}{0.25} + \left(\left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      7. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)} \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      9. distribute-neg-frac 99.9%

         \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}} - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      10. metadata-eval 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \left(-\frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      11. distribute-neg-frac 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      12. metadata-eval 99.9%

          \[\leadsto \frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{\color{blue}{-0.5}}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.125 + {\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}^{3}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. cube-div 99.9%

         \[\leadsto \frac{0.125 + \color{blue}{\frac{{-0.5}^{3}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{0.125 + \frac{\color{blue}{-0.125}}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

      3. distribute-rgt-out-- 99.9%

         \[\leadsto \frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \color{blue}{\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   10. Simplified 99.9%

       \[\leadsto \frac{\color{blue}{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   11. Step-by-step derivation

       1. associate-*l/ 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \color{blue}{\frac{-0.5 \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} - 0.5\right)}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       2. sub-neg 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5 \cdot \color{blue}{\left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + \left(-0.5\right)\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5 \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + \color{blue}{-0.5}\right)}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \color{blue}{\frac{-0.5 \cdot \left(\frac{-0.5}{\mathsf{hypot}\left(1, x\right)} + -0.5\right)}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   13. Step-by-step derivation

       1. +-commutative 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{-0.5 \cdot \color{blue}{\left(-0.5 + \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       2. distribute-lft-in 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{\color{blue}{-0.5 \cdot -0.5 + -0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       3. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{\color{blue}{0.25} + -0.5 \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       4. associate-*r/ 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{0.25 + \color{blue}{\frac{-0.5 \cdot -0.5}{\mathsf{hypot}\left(1, x\right)}}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

       5. metadata-eval 99.9%

          \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{0.25 + \frac{\color{blue}{0.25}}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]

   14. Simplified 99.9%

       \[\leadsto \frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \color{blue}{\frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 + \frac{-0.125}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{3}}}{0.25 + \frac{0.25 + \frac{0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{t\_0}} \cdot \frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0005)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (*
      (/ 1.0 (+ 1.0 (sqrt t_0)))
      (/ (+ 0.25 (/ (/ -0.25 (hypot 1.0 x)) (hypot 1.0 x))) t_0)))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (1.0 / (1.0 + sqrt(t_0))) * ((0.25 + ((-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 + (0.5 / Math.hypot(1.0, x));
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = (1.0 / (1.0 + Math.sqrt(t_0))) * ((0.25 + ((-0.25 / Math.hypot(1.0, x)) / Math.hypot(1.0, x))) / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (0.5 / math.hypot(1.0, x))
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = (1.0 / (1.0 + math.sqrt(t_0))) * ((0.25 + ((-0.25 / math.hypot(1.0, x)) / math.hypot(1.0, x))) / t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + sqrt(t_0))) * Float64(Float64(0.25 + Float64(Float64(-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (0.5 / hypot(1.0, x));
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = (1.0 / (1.0 + sqrt(t_0))) * ((0.25 + ((-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 + N[(N[(-0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. flip-+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.25 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-frac 99.9%

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

      10. metadata-eval 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{t\_0}}{1 + \sqrt{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 (hypot 1.0 x)))))
   (if (<= (hypot 1.0 x) 1.0005)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (/
      (/ (+ 0.25 (/ (/ -0.25 (hypot 1.0 x)) (hypot 1.0 x))) t_0)
      (+ 1.0 (sqrt t_0))))))

C

double code(double x) {
	double t_0 = 0.5 + (0.5 / hypot(1.0, x));
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = ((0.25 + ((-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0) / (1.0 + sqrt(t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 + (0.5 / Math.hypot(1.0, x));
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = ((0.25 + ((-0.25 / Math.hypot(1.0, x)) / Math.hypot(1.0, x))) / t_0) / (1.0 + Math.sqrt(t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 + (0.5 / math.hypot(1.0, x))
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = ((0.25 + ((-0.25 / math.hypot(1.0, x)) / math.hypot(1.0, x))) / t_0) / (1.0 + math.sqrt(t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 + Float64(0.5 / hypot(1.0, x)))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(0.25 + Float64(Float64(-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0) / Float64(1.0 + sqrt(t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 + (0.5 / hypot(1.0, x));
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = ((0.25 + ((-0.25 / hypot(1.0, x)) / hypot(1.0, x))) / t_0) / (1.0 + sqrt(t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.25 + N[(N[(-0.25 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
   7. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. flip-+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. distribute-neg-frac 99.9%

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

      5. metadata-eval 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{\frac{0.25 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)} \cdot \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}{0.5 - \frac{-0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   9. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

      5. distribute-neg-frac 99.9%

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

      6. distribute-neg-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. sub-neg 99.9%

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

      9. distribute-neg-frac 99.9%

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

      10. metadata-eval 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \frac{\color{blue}{\frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25 + \frac{\frac{-0.25}{\mathsf{hypot}\left(1, x\right)}}{\mathsf{hypot}\left(1, x\right)}}{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + t\_0}} \cdot \left(0.5 - t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (* (/ 1.0 (+ 1.0 (sqrt (+ 0.5 t_0)))) (- 0.5 t_0)))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (1.0 / (1.0 + sqrt((0.5 + t_0)))) * (0.5 - t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = (1.0 / (1.0 + Math.sqrt((0.5 + t_0)))) * (0.5 - t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = (1.0 / (1.0 + math.sqrt((0.5 + t_0)))) * (0.5 - t_0)
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(1.0 / Float64(1.0 + sqrt(Float64(0.5 + t_0)))) * Float64(0.5 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = (1.0 / (1.0 + sqrt((0.5 + t_0)))) * (0.5 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. div-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 99.9%

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

      6. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right) \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \cdot \left(0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5}{\mathsf{hypot}\left(1, x\right)}\\ \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - t\_0}{1 + \sqrt{0.5 + t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 1.0005)
     (+
      (* -0.0859375 (pow x 4.0))
      (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
     (/ (- 0.5 t_0) (+ 1.0 (sqrt (+ 0.5 t_0)))))))

C

double code(double x) {
	double t_0 = 0.5 / hypot(1.0, x);
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + ((0.0673828125 * pow(x, 6.0)) + (0.125 * pow(x, 2.0)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.5 / Math.hypot(1.0, x);
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + ((0.0673828125 * Math.pow(x, 6.0)) + (0.125 * Math.pow(x, 2.0)));
	} else {
		tmp = (0.5 - t_0) / (1.0 + Math.sqrt((0.5 + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.5 / math.hypot(1.0, x)
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + ((0.0673828125 * math.pow(x, 6.0)) + (0.125 * math.pow(x, 2.0)))
	else:
		tmp = (0.5 - t_0) / (1.0 + math.sqrt((0.5 + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.5 / hypot(1.0, x))
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	else
		tmp = Float64(Float64(0.5 - t_0) / Float64(1.0 + sqrt(Float64(0.5 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.5 / hypot(1.0, x);
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (0.125 * (x ^ 2.0)));
	else
		tmp = (0.5 - t_0) / (1.0 + sqrt((0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 / N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.4%

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

      2. metadata-eval 98.4%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--r+ 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + \left(0.0673828125 \cdot {x}^{6} + 0.125 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+
    (* -0.0859375 (pow x 4.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* 0.125 (pow x 2.0))))
   (* (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))) (- 0.5 (/ 0.5 x)))))

C

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(Float64(0.0673828125 * (x ^ 6.0)) + Float64(0.125 * (x ^ 2.0))));
	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 / x)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0673828125 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $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 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.3%

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

      2. metadata-eval 53.3%

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

      3. associate-*r/ 53.3%

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

      4. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 99.7%

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

   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. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.5%

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

      2. div-inv 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{x}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{x}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 8: 99.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
   (* (/ 1.0 (+ 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))) (- 0.5 (/ 0.5 x)))))

C

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

Java

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

Python

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

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	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 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = (1.0 / (1.0 + sqrt((0.5 + (0.5 / hypot(1.0, x)))))) * (0.5 - (0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $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 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.3%

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

      2. metadata-eval 53.3%

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

      3. associate-*r/ 53.3%

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

      4. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]

   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. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.5%

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

      2. div-inv 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 99.0%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \frac{1}{x}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.0%

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

      2. metadata-eval 99.0%

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

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\left(0.5 - \frac{0.5}{x}\right)} \cdot \frac{1}{1 + \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

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

Alternative 9: 98.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
   (+ (/ 0.5 (+ 1.0 (sqrt 0.5))) (/ (/ -0.25 x) (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = (0.5 / (1.0 + sqrt(0.5))) + ((-0.25 / x) / sqrt(0.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 2.0) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = (0.5 / (1.0 + Math.sqrt(0.5))) + ((-0.25 / x) / Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = (0.5 / (1.0 + math.sqrt(0.5))) + ((-0.25 / x) / math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(0.5 / Float64(1.0 + sqrt(0.5))) + Float64(Float64(-0.25 / x) / sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = (0.5 / (1.0 + sqrt(0.5))) + ((-0.25 / x) / sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / x), $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.3%

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

      2. metadata-eval 53.3%

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

      3. associate-*r/ 53.3%

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

      4. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]

   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. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 97.4%

      \[\leadsto \color{blue}{1 - \left(\sqrt{0.5} + 0.25 \cdot \frac{1}{x \cdot \sqrt{0.5}}\right)} \]
   6. Step-by-step derivation

      1. associate--r+ 97.4%

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

      2. associate-*r/ 97.4%

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

      3. metadata-eval 97.4%

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

   7. Simplified 97.4%

      \[\leadsto \color{blue}{\left(1 - \sqrt{0.5}\right) - \frac{0.25}{x \cdot \sqrt{0.5}}} \]
   8. Step-by-step derivation

      1. flip-- 97.4%

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5} \cdot \sqrt{0.5}}{1 + \sqrt{0.5}}} - \frac{0.25}{x \cdot \sqrt{0.5}} \]

      2. frac-2neg 97.4%

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

      3. metadata-eval 97.4%

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

      4. rem-square-sqrt 98.9%

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

      5. metadata-eval 98.9%

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

      6. metadata-eval 98.9%

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

   9. Applied egg-rr 98.9%

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

   10. Taylor expanded in x around 0 98.9%

       \[\leadsto \color{blue}{0.5 \cdot \frac{1}{1 + \sqrt{0.5}} - 0.25 \cdot \frac{1}{x \cdot \sqrt{0.5}}} \]
   11. Step-by-step derivation

       1. associate-*r/ 98.9%

          \[\leadsto 0.5 \cdot \frac{1}{1 + \sqrt{0.5}} - \color{blue}{\frac{0.25 \cdot 1}{x \cdot \sqrt{0.5}}} \]

       2. metadata-eval 98.9%

          \[\leadsto 0.5 \cdot \frac{1}{1 + \sqrt{0.5}} - \frac{\color{blue}{0.25}}{x \cdot \sqrt{0.5}} \]

       3. sub-neg 98.9%

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

       4. associate-*r/ 98.9%

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

       5. metadata-eval 98.9%

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

       6. distribute-neg-frac 98.9%

          \[\leadsto \frac{0.5}{1 + \sqrt{0.5}} + \color{blue}{\frac{-0.25}{x \cdot \sqrt{0.5}}} \]

       7. metadata-eval 98.9%

          \[\leadsto \frac{0.5}{1 + \sqrt{0.5}} + \frac{\color{blue}{-0.25}}{x \cdot \sqrt{0.5}} \]

       8. associate-/r* 98.9%

          \[\leadsto \frac{0.5}{1 + \sqrt{0.5}} + \color{blue}{\frac{\frac{-0.25}{x}}{\sqrt{0.5}}} \]

   12. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{0.5}{1 + \sqrt{0.5}} + \frac{\frac{-0.25}{x}}{\sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}} + \frac{\frac{-0.25}{x}}{\sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.0005)
   (+ (* -0.0859375 (pow x 4.0)) (* 0.125 (pow x 2.0)))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * pow(x, 4.0)) + (0.125 * pow(x, 2.0));
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.0005) {
		tmp = (-0.0859375 * Math.pow(x, 4.0)) + (0.125 * Math.pow(x, 2.0));
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.0005:
		tmp = (-0.0859375 * math.pow(x, 4.0)) + (0.125 * math.pow(x, 2.0))
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.0005)
		tmp = Float64(Float64(-0.0859375 * (x ^ 4.0)) + Float64(0.125 * (x ^ 2.0)));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.0005)
		tmp = (-0.0859375 * (x ^ 4.0)) + (0.125 * (x ^ 2.0));
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.0005], N[(N[(-0.0859375 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.125 * N[Power[x, 2.0], $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]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.00049999999999994

   1. Initial program 53.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.1%

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

      2. metadata-eval 53.1%

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

      3. associate-*r/ 53.1%

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

      4. metadata-eval 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]

   if 1.00049999999999994 < (hypot.f64 1 x)

   1. Initial program 98.4%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.4%

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

      2. metadata-eval 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

   3. Simplified 98.4%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.0005:\\ \;\;\;\;-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.000000005:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 1.000000005)
   (* 0.125 (pow x 2.0))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 1.000000005) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 1.000000005) {
		tmp = 0.125 * Math.pow(x, 2.0);
	} else {
		tmp = 1.0 - Math.sqrt((0.5 + (0.5 / Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 1.000000005:
		tmp = 0.125 * math.pow(x, 2.0)
	else:
		tmp = 1.0 - math.sqrt((0.5 + (0.5 / math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 1.000000005)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(1.0 - sqrt(Float64(0.5 + Float64(0.5 / hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 1.000000005)
		tmp = 0.125 * (x ^ 2.0);
	else
		tmp = 1.0 - sqrt((0.5 + (0.5 / hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 1.000000005], N[(0.125 * N[Power[x, 2.0], $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]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 1.000000005

   1. Initial program 53.0%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.0%

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

      2. metadata-eval 53.0%

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

      3. associate-*r/ 53.0%

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

      4. metadata-eval 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   if 1.000000005 < (hypot.f64 1 x)

   1. Initial program 98.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.1%

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

      2. metadata-eval 98.1%

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

      3. associate-*r/ 98.1%

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

      4. metadata-eval 98.1%

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

   3. Simplified 98.1%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 1.000000005:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0) (* 0.125 (pow x 2.0)) (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = 0.5 / (1.0 + sqrt(0.5));
	}
	return tmp;
}

Java

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

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = 0.125 * math.pow(x, 2.0)
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(0.5 / Float64(1.0 + sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (hypot(1.0, x) <= 2.0)
		tmp = 0.125 * (x ^ 2.0);
	else
		tmp = 0.5 / (1.0 + sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 2.0], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(1.0 + N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 53.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 53.3%

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

      2. metadata-eval 53.3%

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

      3. associate-*r/ 53.3%

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

      4. metadata-eval 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 99.0%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   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. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

      \[\leadsto \color{blue}{1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip-- 98.5%

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

      2. div-inv 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{\frac{0.5}{1 + \sqrt{0.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55) (* 0.125 (pow x 2.0)) (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * pow(x, 2.0);
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = 0.125d0 * (x ** 2.0d0)
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = 0.125 * Math.pow(x, 2.0);
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = 0.125 * math.pow(x, 2.0)
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(0.125 * (x ^ 2.0));
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = 0.125 * (x ^ 2.0);
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(0.125 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 68.2%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 68.2%

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

      2. metadata-eval 68.2%

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

      3. associate-*r/ 68.2%

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

      4. metadata-eval 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in x around 0 67.7%

      \[\leadsto \color{blue}{0.125 \cdot {x}^{2}} \]

   if 1.55000000000000004 < x

   1. Initial program 98.5%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-*r/ 98.5%

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

      4. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in x around inf 96.3%

      \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;0.125 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.15e-77) 0.0 (- 1.0 (sqrt 0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - sqrt(0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.15d-77) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 - sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.15e-77) {
		tmp = 0.0;
	} else {
		tmp = 1.0 - Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.15e-77:
		tmp = 0.0
	else:
		tmp = 1.0 - math.sqrt(0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = Float64(1.0 - sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.15e-77)
		tmp = 0.0;
	else
		tmp = 1.0 - sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.15e-77], 0.0, N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1500000000000001e-77

   1. Initial program 74.3%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 74.3%

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

      2. metadata-eval 74.3%

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

      3. associate-*r/ 74.3%

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

      4. metadata-eval 74.3%

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

   3. Simplified 74.3%

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

   5. Taylor expanded in x around 0 39.1%

      \[\leadsto 1 - \color{blue}{1} \]

   if 2.1500000000000001e-77 < x

   1. Initial program 81.1%

      \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
   2. Step-by-step derivation

      1. distribute-lft-in 81.1%

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

      2. metadata-eval 81.1%

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

      3. associate-*r/ 81.1%

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

      4. metadata-eval 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 79.0%

      \[\leadsto \color{blue}{1 - \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.5% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 76.6%

   \[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \]
2. Step-by-step derivation

   1. distribute-lft-in 76.6%

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

   2. metadata-eval 76.6%

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

   3. associate-*r/ 76.6%

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

   4. metadata-eval 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in x around 0 26.8%

   \[\leadsto 1 - \color{blue}{1} \]

6. Final simplification 26.8%

   \[\leadsto 0 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))