Given's Rotation SVD example, simplified

Percentage Accurate: 75.6% → 99.8%

Time: 8.6s

Alternatives: 16

Speedup: 2.0×

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: 75.6% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 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 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) <= 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 + 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) <= 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 + 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) <= 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 + 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) <= 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(1.0 / Float64(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) <= 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 + 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], 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[(1.0 / N[(N[(1.0 + N[Sqrt[N[(0.5 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. 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 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. 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. clear-num 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 0.5 (hypot 1.0 x))))
   (if (<= (hypot 1.0 x) 2.0)
     (+
      (* -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) <= 2.0) {
		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) <= 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 = (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) <= 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 = (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) <= 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(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) <= 2.0)
		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], 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[(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) < 2

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. 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 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. 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. metadata-eval 98.5%

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

      3. add-sqr-sqrt 100.0%

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

      4. associate--r+ 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (+
    (* -0.0859375 (pow x 4.0))
    (+ (* 0.0673828125 (pow x 6.0)) (* 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) <= 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 - 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) <= 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 - 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) <= 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 - 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) <= 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(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) <= 2.0)
		tmp = (-0.0859375 * (x ^ 4.0)) + ((0.0673828125 * (x ^ 6.0)) + (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], 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[(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) < 2

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. 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 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%

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

4. Final simplification 99.3%

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

Alternative 4: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 (hypot.f64 1 x)) < 0.5

   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%

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

   if 0.5 < (/.f64 1 (hypot.f64 1 x))

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-0.0859375 \cdot {x}^{4} + 0.125 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

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

Alternative 5: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \left({x}^{2} \cdot -0.53125 + \frac{\frac{8}{x}}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (/ 1.0 (+ 5.5 (+ (* (pow x 2.0) -0.53125) (/ (/ 8.0 x) x))))
   (- 1.0 (sqrt (+ 0.5 (/ 0.5 (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 1.0 / (5.5 + ((pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)));
	} 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) <= 2.0) {
		tmp = 1.0 / (5.5 + ((Math.pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)));
	} 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) <= 2.0:
		tmp = 1.0 / (5.5 + ((math.pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)))
	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) <= 2.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(Float64((x ^ 2.0) * -0.53125) + Float64(Float64(8.0 / x) / x))));
	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) <= 2.0)
		tmp = 1.0 / (5.5 + (((x ^ 2.0) * -0.53125) + ((8.0 / x) / x)));
	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], 2.0], N[(1.0 / N[(5.5 + N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.53125), $MachinePrecision] + N[(N[(8.0 / x), $MachinePrecision] / x), $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]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

      1. flip-- 55.0%

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

      2. clear-num 55.0%

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

      3. metadata-eval 55.0%

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

      4. add-sqr-sqrt 55.0%

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

      5. associate--r+ 55.0%

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

      6. metadata-eval 55.0%

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

   5. Applied egg-rr 55.0%

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

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \left(-0.53125 \cdot {x}^{2} + 8 \cdot \frac{1}{{x}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 99.4%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{8}{{x}^{2}}}\right)} \]

      2. unpow2 99.4%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \frac{8}{\color{blue}{x \cdot x}}\right)} \]

      3. associate-/r* 99.3%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{\frac{8}{x}}{x}}\right)} \]

   8. Applied egg-rr 99.3%

      \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{\frac{8}{x}}{x}}\right)} \]

   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%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \left({x}^{2} \cdot -0.53125 + \frac{\frac{8}{x}}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{0.5 + \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}\\ \end{array} \]

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \left({x}^{2} \cdot -0.53125 + \frac{\frac{8}{x}}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (/ 1.0 (+ 5.5 (+ (* (pow x 2.0) -0.53125) (/ (/ 8.0 x) x))))
   (/ (- 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 1.0 / (5.5 + ((pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((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 = 1.0 / (5.5 + ((Math.pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = 1.0 / (5.5 + ((math.pow(x, 2.0) * -0.53125) + ((8.0 / x) / x)))
	else:
		tmp = (0.5 - (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (-0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(Float64((x ^ 2.0) * -0.53125) + Float64(Float64(8.0 / x) / x))));
	else
		tmp = Float64(Float64(0.5 - Float64(-0.5 / x)) / Float64(1.0 + sqrt(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 = 1.0 / (5.5 + (((x ^ 2.0) * -0.53125) + ((8.0 / x) / x)));
	else
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((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[(1.0 / N[(5.5 + N[(N[(N[Power[x, 2.0], $MachinePrecision] * -0.53125), $MachinePrecision] + N[(N[(8.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

      1. flip-- 55.0%

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

      2. clear-num 55.0%

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

      3. metadata-eval 55.0%

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

      4. add-sqr-sqrt 55.0%

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

      5. associate--r+ 55.0%

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

      6. metadata-eval 55.0%

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

   5. Applied egg-rr 55.0%

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

   6. Taylor expanded in x around 0 99.4%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \left(-0.53125 \cdot {x}^{2} + 8 \cdot \frac{1}{{x}^{2}}\right)}} \]
   7. Step-by-step derivation

      1. div-inv 99.4%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{8}{{x}^{2}}}\right)} \]

      2. unpow2 99.4%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \frac{8}{\color{blue}{x \cdot x}}\right)} \]

      3. associate-/r* 99.3%

         \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{\frac{8}{x}}{x}}\right)} \]

   8. Applied egg-rr 99.3%

      \[\leadsto \frac{1}{5.5 + \left(-0.53125 \cdot {x}^{2} + \color{blue}{\frac{\frac{8}{x}}{x}}\right)} \]

   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. Taylor expanded in x around -inf 96.7%

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

      1. flip-- 96.7%

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

      2. metadata-eval 96.7%

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

      3. add-sqr-sqrt 98.1%

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

      4. associate--r+ 98.1%

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

      5. metadata-eval 98.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \left({x}^{2} \cdot -0.53125 + \frac{\frac{8}{x}}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \end{array} \]

Alternative 7: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 4.0)
   (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x)))))
   (/ (+ 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 + (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 4.0:
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)))
	else:
		tmp = (0.5 + (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 4.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))));
	else
		tmp = Float64(Float64(0.5 + Float64(-0.5 / x)) / Float64(1.0 + sqrt(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) <= 4.0)
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	else
		tmp = (0.5 + (-0.5 / x)) / (1.0 + sqrt((0.5 + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 4.0], N[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 4

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

      1. flip-- 55.3%

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

      2. clear-num 55.3%

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

      3. metadata-eval 55.3%

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

      4. add-sqr-sqrt 55.3%

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

      5. associate--r+ 55.3%

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

      6. metadata-eval 55.3%

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

   5. Applied egg-rr 55.3%

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

   6. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

      2. metadata-eval 98.5%

         \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   8. Simplified 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

      3. inv-pow 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

      4. unpow2 98.5%

         \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

      5. unpow-prod-down 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

      6. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

      7. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

      8. associate-*l* 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

   10. Applied egg-rr 98.3%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.3%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       2. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       3. associate-*l/ 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

       4. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

       5. frac-times 98.5%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

       6. metadata-eval 98.5%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]

   if 4 < (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. Taylor expanded in x around inf 96.5%

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

   6. Applied egg-rr 98.0%

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

      1. distribute-neg-in 98.0%

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

      2. metadata-eval 98.0%

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

      3. remove-double-neg 98.0%

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

      4. distribute-neg-in 98.0%

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

      5. metadata-eval 98.0%

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

      6. distribute-neg-frac 98.0%

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

      7. metadata-eval 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{0.5}{x}}}\\ \end{array} \]

Alternative 8: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 4.0)
   (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x)))))
   (/ (- 0.5 (/ -0.5 x)) (+ 1.0 (sqrt (+ 0.5 (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.hypot(1.0, x) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + Math.sqrt((0.5 + (-0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 4.0:
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)))
	else:
		tmp = (0.5 - (-0.5 / x)) / (1.0 + math.sqrt((0.5 + (-0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 4.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))));
	else
		tmp = Float64(Float64(0.5 - Float64(-0.5 / x)) / Float64(1.0 + sqrt(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) <= 4.0)
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	else
		tmp = (0.5 - (-0.5 / x)) / (1.0 + sqrt((0.5 + (-0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision], 4.0], N[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Sqrt[N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (hypot.f64 1 x) < 4

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

      1. flip-- 55.3%

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

      2. clear-num 55.3%

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

      3. metadata-eval 55.3%

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

      4. add-sqr-sqrt 55.3%

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

      5. associate--r+ 55.3%

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

      6. metadata-eval 55.3%

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

   5. Applied egg-rr 55.3%

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

   6. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

      2. metadata-eval 98.5%

         \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   8. Simplified 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

      3. inv-pow 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

      4. unpow2 98.5%

         \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

      5. unpow-prod-down 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

      6. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

      7. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

      8. associate-*l* 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

   10. Applied egg-rr 98.3%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.3%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       2. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       3. associate-*l/ 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

       4. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

       5. frac-times 98.5%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

       6. metadata-eval 98.5%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]

   if 4 < (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. Taylor expanded in x around -inf 97.3%

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

      1. flip-- 97.3%

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

      2. metadata-eval 97.3%

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

      3. add-sqr-sqrt 98.7%

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

      4. associate--r+ 98.7%

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

      5. metadata-eval 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5 + \frac{-0.5}{x}}}\\ \end{array} \]

Alternative 9: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 2.0)
   (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x)))))
   (/ (- 0.5 (/ -0.5 x)) (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 2.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (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 = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = (0.5 - (-0.5 / x)) / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 2.0:
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)))
	else:
		tmp = (0.5 - (-0.5 / x)) / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 2.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))));
	else
		tmp = Float64(Float64(0.5 - Float64(-0.5 / x)) / 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 = 1.0 / (5.5 + (-8.0 / (x * -x)));
	else
		tmp = (0.5 - (-0.5 / x)) / (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[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / 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 55.0%

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

      1. distribute-lft-in 55.0%

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

      2. metadata-eval 55.0%

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

      3. associate-*r/ 55.0%

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

      4. metadata-eval 55.0%

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

   3. Simplified 55.0%

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

      1. flip-- 55.0%

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

      2. clear-num 55.0%

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

      3. metadata-eval 55.0%

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

      4. add-sqr-sqrt 55.0%

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

      5. associate--r+ 55.0%

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

      6. metadata-eval 55.0%

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

   5. Applied egg-rr 55.0%

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

   6. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 99.1%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

      2. metadata-eval 99.1%

         \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   8. Simplified 99.1%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 99.1%

         \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

      3. inv-pow 99.1%

         \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

      4. unpow2 99.1%

         \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

      5. unpow-prod-down 98.8%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

      6. inv-pow 98.8%

         \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

      7. inv-pow 98.8%

         \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

      8. associate-*l* 98.8%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

   10. Applied egg-rr 98.8%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.8%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       2. metadata-eval 98.8%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       3. associate-*l/ 98.8%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

       4. metadata-eval 98.8%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

       5. frac-times 99.1%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

       6. metadata-eval 99.1%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

   12. Applied egg-rr 99.1%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]

   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. Taylor expanded in x around -inf 96.7%

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

      1. flip-- 96.7%

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

      2. metadata-eval 96.7%

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

      3. add-sqr-sqrt 98.1%

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

      4. associate--r+ 98.1%

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

      5. metadata-eval 98.1%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in x around inf 97.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 2:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 - \frac{-0.5}{x}}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 10: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (hypot 1.0 x) 4.0)
   (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x)))))
   (/ 0.5 (+ 1.0 (sqrt 0.5)))))

C

double code(double x) {
	double tmp;
	if (hypot(1.0, x) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} 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) <= 4.0) {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	} else {
		tmp = 0.5 / (1.0 + Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.hypot(1.0, x) <= 4.0:
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)))
	else:
		tmp = 0.5 / (1.0 + math.sqrt(0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (hypot(1.0, x) <= 4.0)
		tmp = Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))));
	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) <= 4.0)
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	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], 4.0], N[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $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) < 4

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

      1. flip-- 55.3%

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

      2. clear-num 55.3%

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

      3. metadata-eval 55.3%

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

      4. add-sqr-sqrt 55.3%

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

      5. associate--r+ 55.3%

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

      6. metadata-eval 55.3%

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

   5. Applied egg-rr 55.3%

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

   6. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

      2. metadata-eval 98.5%

         \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   8. Simplified 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

      3. inv-pow 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

      4. unpow2 98.5%

         \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

      5. unpow-prod-down 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

      6. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

      7. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

      8. associate-*l* 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

   10. Applied egg-rr 98.3%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.3%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       2. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       3. associate-*l/ 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

       4. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

       5. frac-times 98.5%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

       6. metadata-eval 98.5%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]

   if 4 < (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. 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. clear-num 98.5%

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

      3. metadata-eval 98.5%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--r+ 100.0%

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

      6. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 97.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{hypot}\left(1, x\right) \leq 4:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{1 + \sqrt{0.5}}\\ \end{array} \]

Alternative 11: 97.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \lor \neg \left(x \leq 4.6\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.6) (not (<= x 4.6)))
   (- 1.0 (sqrt 0.5))
   (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x)))))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.6) || !(x <= 4.6)) {
		tmp = 1.0 - sqrt(0.5);
	} else {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.6d0)) .or. (.not. (x <= 4.6d0))) then
        tmp = 1.0d0 - sqrt(0.5d0)
    else
        tmp = 1.0d0 / (5.5d0 + ((-8.0d0) / (x * -x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.6) || !(x <= 4.6)) {
		tmp = 1.0 - Math.sqrt(0.5);
	} else {
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.6) or not (x <= 4.6):
		tmp = 1.0 - math.sqrt(0.5)
	else:
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.6) || !(x <= 4.6))
		tmp = Float64(1.0 - sqrt(0.5));
	else
		tmp = Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.6) || ~((x <= 4.6)))
		tmp = 1.0 - sqrt(0.5);
	else
		tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.6], N[Not[LessEqual[x, 4.6]], $MachinePrecision]], N[(1.0 - N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5999999999999996 or 4.5999999999999996 < 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. Taylor expanded in x around inf 95.9%

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

   if -4.5999999999999996 < x < 4.5999999999999996

   1. Initial program 55.3%

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

      1. distribute-lft-in 55.3%

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

      2. metadata-eval 55.3%

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

      3. associate-*r/ 55.3%

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

      4. metadata-eval 55.3%

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

   3. Simplified 55.3%

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

      1. flip-- 55.3%

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

      2. clear-num 55.3%

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

      3. metadata-eval 55.3%

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

      4. add-sqr-sqrt 55.3%

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

      5. associate--r+ 55.3%

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

      6. metadata-eval 55.3%

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

   5. Applied egg-rr 55.3%

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

   6. Taylor expanded in x around 0 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

      2. metadata-eval 98.5%

         \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

   8. Simplified 98.5%

      \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. div-inv 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

      2. *-commutative 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

      3. inv-pow 98.5%

         \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

      4. unpow2 98.5%

         \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

      5. unpow-prod-down 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

      6. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

      7. inv-pow 98.3%

         \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

      8. associate-*l* 98.3%

         \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

   10. Applied egg-rr 98.3%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
   11. Step-by-step derivation

       1. frac-2neg 98.3%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       2. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

       3. associate-*l/ 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

       4. metadata-eval 98.3%

          \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

       5. frac-times 98.5%

          \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

       6. metadata-eval 98.5%

          \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \lor \neg \left(x \leq 4.6\right):\\ \;\;\;\;1 - \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}}\\ \end{array} \]

Alternative 12: 58.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 5.5 (/ -8.0 (* x (- x))))))

C

double code(double x) {
	return 1.0 / (5.5 + (-8.0 / (x * -x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (5.5d0 + ((-8.0d0) / (x * -x)))
end function

Java

public static double code(double x) {
	return 1.0 / (5.5 + (-8.0 / (x * -x)));
}

Python

def code(x):
	return 1.0 / (5.5 + (-8.0 / (x * -x)))

Julia

function code(x)
	return Float64(1.0 / Float64(5.5 + Float64(-8.0 / Float64(x * Float64(-x)))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (5.5 + (-8.0 / (x * -x)));
end

Wolfram

code[x_] := N[(1.0 / N[(5.5 + N[(-8.0 / N[(x * (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. distribute-lft-in 76.2%

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

   2. metadata-eval 76.2%

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

   3. associate-*r/ 76.2%

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

   4. metadata-eval 76.2%

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

3. Simplified 76.2%

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

   1. flip-- 76.2%

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

   2. clear-num 76.2%

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

   3. metadata-eval 76.2%

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

   4. add-sqr-sqrt 76.9%

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

   5. associate--r+ 77.0%

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

   6. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

6. Taylor expanded in x around 0 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
7. Step-by-step derivation

   1. associate-*r/ 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 60.3%

      \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

8. Simplified 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
9. Step-by-step derivation

   1. div-inv 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

   2. *-commutative 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

   3. inv-pow 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

   4. unpow2 60.3%

      \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

   5. unpow-prod-down 60.1%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

   6. inv-pow 60.1%

      \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

   7. inv-pow 60.1%

      \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

   8. associate-*l* 60.1%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

10. Applied egg-rr 60.1%

    \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
11. Step-by-step derivation

    1. frac-2neg 60.1%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1}{-x}} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

    2. metadata-eval 60.1%

       \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-1}}{-x} \cdot \left(\frac{1}{x} \cdot 8\right)} \]

    3. associate-*l/ 60.1%

       \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \color{blue}{\frac{1 \cdot 8}{x}}} \]

    4. metadata-eval 60.1%

       \[\leadsto \frac{1}{5.5 + \frac{-1}{-x} \cdot \frac{\color{blue}{8}}{x}} \]

    5. frac-times 60.3%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-1 \cdot 8}{\left(-x\right) \cdot x}}} \]

    6. metadata-eval 60.3%

       \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{-8}}{\left(-x\right) \cdot x}} \]

12. Applied egg-rr 60.3%

    \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{-8}{\left(-x\right) \cdot x}}} \]

13. Final simplification 60.3%

    \[\leadsto \frac{1}{5.5 + \frac{-8}{x \cdot \left(-x\right)}} \]

Alternative 13: 58.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{5.5 + \frac{\frac{8}{x}}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 5.5 (/ (/ 8.0 x) x))))

C

double code(double x) {
	return 1.0 / (5.5 + ((8.0 / x) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (5.5d0 + ((8.0d0 / x) / x))
end function

Java

public static double code(double x) {
	return 1.0 / (5.5 + ((8.0 / x) / x));
}

Python

def code(x):
	return 1.0 / (5.5 + ((8.0 / x) / x))

Julia

function code(x)
	return Float64(1.0 / Float64(5.5 + Float64(Float64(8.0 / x) / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (5.5 + ((8.0 / x) / x));
end

Wolfram

code[x_] := N[(1.0 / N[(5.5 + N[(N[(8.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

   1. distribute-lft-in 76.2%

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

   2. metadata-eval 76.2%

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

   3. associate-*r/ 76.2%

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

   4. metadata-eval 76.2%

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

3. Simplified 76.2%

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

   1. flip-- 76.2%

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

   2. clear-num 76.2%

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

   3. metadata-eval 76.2%

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

   4. add-sqr-sqrt 76.9%

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

   5. associate--r+ 77.0%

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

   6. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

6. Taylor expanded in x around 0 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
7. Step-by-step derivation

   1. associate-*r/ 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 60.3%

      \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

8. Simplified 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]
9. Step-by-step derivation

   1. div-inv 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{8 \cdot \frac{1}{{x}^{2}}}} \]

   2. *-commutative 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{{x}^{2}} \cdot 8}} \]

   3. inv-pow 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{{\left({x}^{2}\right)}^{-1}} \cdot 8} \]

   4. unpow2 60.3%

      \[\leadsto \frac{1}{5.5 + {\color{blue}{\left(x \cdot x\right)}}^{-1} \cdot 8} \]

   5. unpow-prod-down 60.1%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot 8} \]

   6. inv-pow 60.1%

      \[\leadsto \frac{1}{5.5 + \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot 8} \]

   7. inv-pow 60.1%

      \[\leadsto \frac{1}{5.5 + \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot 8} \]

   8. associate-*l* 60.1%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]

10. Applied egg-rr 60.1%

    \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot 8\right)}} \]
11. Step-by-step derivation

    1. associate-*l/ 60.2%

       \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{1 \cdot \left(\frac{1}{x} \cdot 8\right)}{x}}} \]

    2. *-un-lft-identity 60.2%

       \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{\frac{1}{x} \cdot 8}}{x}} \]

    3. associate-*l/ 60.2%

       \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{\frac{1 \cdot 8}{x}}}{x}} \]

    4. metadata-eval 60.2%

       \[\leadsto \frac{1}{5.5 + \frac{\frac{\color{blue}{8}}{x}}{x}} \]

12. Applied egg-rr 60.2%

    \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{\frac{8}{x}}{x}}} \]

13. Final simplification 60.2%

    \[\leadsto \frac{1}{5.5 + \frac{\frac{8}{x}}{x}} \]

Alternative 14: 36.9% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.1e-77) 0.25 (if (<= x 2.1e-77) 0.0 0.25)))

C

double code(double x) {
	double tmp;
	if (x <= -2.1e-77) {
		tmp = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2.1d-77)) then
        tmp = 0.25d0
    else if (x <= 2.1d-77) then
        tmp = 0.0d0
    else
        tmp = 0.25d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.1e-77) {
		tmp = 0.25;
	} else if (x <= 2.1e-77) {
		tmp = 0.0;
	} else {
		tmp = 0.25;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.1e-77:
		tmp = 0.25
	elif x <= 2.1e-77:
		tmp = 0.0
	else:
		tmp = 0.25
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.1e-77)
		tmp = 0.25;
	elseif (x <= 2.1e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.1e-77)
		tmp = 0.25;
	elseif (x <= 2.1e-77)
		tmp = 0.0;
	else
		tmp = 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.1e-77], 0.25, If[LessEqual[x, 2.1e-77], 0.0, 0.25]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000015e-77 or 2.10000000000000015e-77 < x

   1. Initial program 80.9%

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

      1. distribute-lft-in 80.9%

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

      2. metadata-eval 80.9%

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

      3. associate-*r/ 80.9%

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

      4. metadata-eval 80.9%

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

   3. Simplified 80.9%

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

      1. flip-- 80.9%

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

      2. metadata-eval 80.9%

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

      3. add-sqr-sqrt 82.1%

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

      4. associate--r+ 82.1%

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

      5. metadata-eval 82.1%

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

   5. Applied egg-rr 82.1%

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

   6. Taylor expanded in x around 0 20.0%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

   7. Taylor expanded in x around inf 19.7%

      \[\leadsto \frac{\color{blue}{0.5}}{2} \]

   if -2.10000000000000015e-77 < x < 2.10000000000000015e-77

   1. Initial program 68.9%

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

      1. distribute-lft-in 68.9%

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

      2. metadata-eval 68.9%

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

      3. associate-*r/ 68.9%

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

      4. metadata-eval 68.9%

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

   3. Simplified 68.9%

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

      1. flip-- 68.9%

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

      2. metadata-eval 68.9%

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

      3. add-sqr-sqrt 68.9%

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

      4. associate--r+ 68.9%

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

      5. metadata-eval 68.9%

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

   5. Applied egg-rr 68.9%

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

   6. Taylor expanded in x around 0 68.9%

      \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

   7. Taylor expanded in x around 0 68.9%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-77}:\\ \;\;\;\;0.25\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-77}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.25\\ \end{array} \]

Alternative 15: 12.0% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.18181818181818182 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.18181818181818182)

C

double code(double x) {
	return 0.18181818181818182;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.18181818181818182d0
end function

Java

public static double code(double x) {
	return 0.18181818181818182;
}

Python

def code(x):
	return 0.18181818181818182

Julia

function code(x)
	return 0.18181818181818182
end

MATLAB

function tmp = code(x)
	tmp = 0.18181818181818182;
end

Wolfram

code[x_] := 0.18181818181818182

Derivation

1. Initial program 76.2%

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

   1. distribute-lft-in 76.2%

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

   2. metadata-eval 76.2%

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

   3. associate-*r/ 76.2%

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

   4. metadata-eval 76.2%

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

3. Simplified 76.2%

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

   1. flip-- 76.2%

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

   2. clear-num 76.2%

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

   3. metadata-eval 76.2%

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

   4. add-sqr-sqrt 76.9%

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

   5. associate--r+ 77.0%

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

   6. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

6. Taylor expanded in x around 0 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + 8 \cdot \frac{1}{{x}^{2}}}} \]
7. Step-by-step derivation

   1. associate-*r/ 60.3%

      \[\leadsto \frac{1}{5.5 + \color{blue}{\frac{8 \cdot 1}{{x}^{2}}}} \]

   2. metadata-eval 60.3%

      \[\leadsto \frac{1}{5.5 + \frac{\color{blue}{8}}{{x}^{2}}} \]

8. Simplified 60.3%

   \[\leadsto \frac{1}{\color{blue}{5.5 + \frac{8}{{x}^{2}}}} \]

9. Taylor expanded in x around inf 11.7%

   \[\leadsto \color{blue}{0.18181818181818182} \]

10. Final simplification 11.7%

    \[\leadsto 0.18181818181818182 \]

Alternative 16: 13.6% accurate, 210.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.25)

C

double code(double x) {
	return 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.25d0
end function

Java

public static double code(double x) {
	return 0.25;
}

Python

def code(x):
	return 0.25

Julia

function code(x)
	return 0.25
end

MATLAB

function tmp = code(x)
	tmp = 0.25;
end

Wolfram

code[x_] := 0.25

Derivation

1. Initial program 76.2%

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

   1. distribute-lft-in 76.2%

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

   2. metadata-eval 76.2%

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

   3. associate-*r/ 76.2%

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

   4. metadata-eval 76.2%

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

3. Simplified 76.2%

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

   1. flip-- 76.2%

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

   2. metadata-eval 76.2%

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

   3. add-sqr-sqrt 77.0%

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

   4. associate--r+ 77.0%

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

   5. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

6. Taylor expanded in x around 0 39.1%

   \[\leadsto \frac{0.5 - \frac{0.5}{\mathsf{hypot}\left(1, x\right)}}{\color{blue}{2}} \]

7. Taylor expanded in x around inf 13.3%

   \[\leadsto \frac{\color{blue}{0.5}}{2} \]

8. Final simplification 13.3%

   \[\leadsto 0.25 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "Given's Rotation SVD example, simplified"
  :precision binary64
  (- 1.0 (sqrt (* 0.5 (+ 1.0 (/ 1.0 (hypot 1.0 x)))))))